# 5.4: Classifying Finite Groups - Mathematics

We've seen that group theory can't distinguish between groups that are isomorphic. So a natural question is whether we can make a list of all of the groups!

We can make new groups from old groups using the direct product. So it would be nice to focus on groups that are not direct products. In the commutative case, this turns out to be pretty straightforward: a (finite) commutative group is a direct product of subgroups if and only if it has a proper subgroup.

The non-commutative case is much more difficult, though. There are actually a few other ways to build new groups from old groups; the most important of these other ways is the semi-direct product; we won't describe how to build semi-direct products here, but you can read about them elsewhere. Importantly, one can 'undo' a semi-direct product using a quotient, the same way one can undo a direct product. To get a sense of how useful the construction is, the symmetric group (S_n) is the semi-direct product of (A_n) and (mathbb{Z}_2). Also, the dihedral group (D_n) is a semi-direct product of (mathbb{Z}_n) and (mathbb{Z}_2).

An interesting question, then, is 'Which groups have no quotients?' We've seen that we can form a quotient group whenever there is a normal subgroup.

Definition 5.3.0: Simple Groups

A group is simple if it has no proper normal subgroups. (A proper subgroup is any subgroup of (G) that is not equal to (G) or ({1}), which are always normal subgroups.)

We'll now actually classify all of the finite simple groups, and discuss some of the history of the non-commutative case.

## The Commutative Case

We can actually classify all of the finite commutative groups pretty easily. First, recall that every subgroup of a commutative group is normal.

Proposition 5.3.1

A finite commutative group is simple if and only if it has prime order (p). In this case, it is isomorphic to the cyclic group, (mathbb{Z}_p).

Proof 5.3.2

If a finite commutative group has prime order then it has no proper subgroups, by Lagrange's theorem. Then it must be simple.

For the other direction, we assume (G) is a finite commutative simple group. (G) must be cyclic, or else we could form a proper subgroup by taking powers of a generator. So (Gsim mathbb{Z}_n) for some (n). But if (n) is not prime we can find a subgroup using a proper divisor of (n). Then (Gsim mathbb{Z}_p) for some prime (p).

Theorem 5.3.3

Every finite commutative group is a direct product of cyclic groups of prime order.

Proof 5.3.4

Let (A) be a commutative group with (n) elements. Take any element (x) not equal to the identity in (A); we know that there is some minimal integer (m) for which (x^m=1). Then (A) has a subgroup of order (m) generated by (x), isomorphic to (mathbb{Z}_m). As a result, we have (Asim A_1 otimes mathbb{Z}_m), where (A_1) is the quotient (A/ mathord mathbb{Z}_m).

We can repeat that procedure indefinitely (taking an (x) in (A_1) and writing (A_1) as a product, and so on), until we obtain a decomposition (A=mathbb{Z}_{m_1}otimes mathbb{Z}_{m_k}), a product of cyclic groups.

We can then use the same trick to decompose each (mathbb{Z}_m) into a direct product of cyclic groups of prime order, completing the proof.

One can extend this trick to some infinite groups: those which have a finite number of generators. (Such groups, unsurprisingly, are called finitely-generated.) This gives rise to the Fundamental theorem of finitely-generated commutative groups.

Suppose (A) is a finitely generated commutative group with infinite cardinality. Show that (Asim mathbb{Z} otimes A'), where (A') is a finitely-generated commutative group.

## The Non-Commutative Case

One of the major projects of 20th century mathematics research was to classify all of the finite simple groups; the project took fifty years, and the proof of the classification is estimated to span 10,000 pages written by over 100 authors. There's currently an effort underway to simplify the proof, however.

The classification shows that all finite simple groups are of one of four types:

1. Commutative groups of prime order,
2. Alternating groups (A_n) with (ngeq 5),
3. Groups of Lie type,
4. The 26 sporadic groups.

We've already seen the first two types of simple group. It turns out that 'most' finite simple groups are in the third class, groups of Lie type, which are well beyond the scope of these notes to construct. Basically, though, groups of Lie type are certain groups of matrices with entries from a finite field, which are we'll see in the next chapter. The 'sporadic' groups are just those groups that don't fit into any of the other three classes!

## 5.4: Classifying Finite Groups - Mathematics

1) Suppose we select 3 cards from a standard deck of cards. Find the probabilities below. We will assume each simple event (each 3-card hand) is equally likely to occur and we know n(S)=52C3= 22,100.

a) Event of getting two face cards and one nonface card.

b) Event of getting all Kings

c) Event of getting 2 Kings and a Jack.

d) Event of 2 hearts and 1 diamond.

2) Suppose we have a basket of 7 good apples and 5 bad apples. If we randomly select 4 apples, what is the probability of getting,

a) exactly 3 good apples? (3 good, 1 bad)

b) exactly 4 good apples? (4 good, 0 bad)

c) no good apples? (0 good, 4 bad)

d) exactly 5 good apples

e) at least 3 good apples (3 good, 1 bad) or (4 good, 0 bad)

3) A committee of 4 people is to be chosen from a group of 5 men and 6 women. What is the probability that the committee will consist of 2 men and 2 women?

4) Suppose your class has 50 students. There are 23 students majoring in psychology and 16 majoring in sociology. Seven students are seeking a double major in psychology and sociology. If a student is randomly selected, what is the probability that you pick a student,

a) majoring in psychology alone?

b) majoring in sociology alone?

c) majoring in EITHER psychology or sociology?

d) majoring in NEITHER psychology nor sociology?

e) NOT majoring in psychology

5) A department store receives a shipment of 27 new portable radios. There are 4 defective radios in the shipment. If 6 radios are selected for display, what is the probability that 2 of them are defective?

6) Eight cards are drawn from a standard deck of cards. What is the probability that there are 4 face cards and 4 non-face cards?

7) A committee of 6 people is to be chosen from a group of 6 men and 5 women. What is the probability that the committee will consist of at least 4 men?

8) A poll was conducted preceding an election to determine the relationship between voter persuasion concerning a controversial issue and the area of the city in which the voter lives. Five hundred registered voters were interviewed from three areas of the city. The data are shown below. Compute the probability of

## Classification of finite simple groups

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four classes described below. These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference with respect to the case of integer factorization is that such "building blocks" do not necessarily determine uniquely a group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

The proof of the classification theorem consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Gorenstein (d.1992), Lyons, and Solomon are gradually publishing a simplified and revised version of the proof.

Statement of the classification theorem
Main article: List of finite simple groups

Theorem — Every finite simple group is isomorphic to one of the following groups:

Theorem — Every finite simple group is isomorphic to one of the following groups:

• A cyclic group with prime order
• An alternating group of degree at least 5
• A simple group of Lie type, including both
• the classical Lie groups, namely the simple groups related to the projective special linear, unitary, symplectic, or orthogonal transformations over a finite field
• the exceptional and twisted groups of Lie type (including the Tits group).

The classification theorem has applications in many branches of mathematics, as questions about the structure of finite groups (and their action on other mathematical objects) can sometimes be reduced to questions about finite simple groups. Thanks to the classification theorem, such questions can sometimes be answered by checking each family of simple groups and each sporadic group.

Daniel Gorenstein announced in 1983 that the finite simple groups had all been classified, but this was premature as he had been misinformed about the proof of the classification of quasithin groups. The completed proof of the classification was announced by Aschbacher (2004) after Aschbacher and Smith published a 1221 page proof for the missing quasithin case.
Overview of the proof of the classification theorem

Gorenstein (1982, 1983) wrote two volumes outlining the low rank and odd characteristic part of the proof, and Michael Aschbacher, Richard Lyons, and Stephen D. Smith et al. (2011) wrote a 3rd volume covering the remaining characteristic 2 case. The proof can be broken up into several major pieces as follows:
Groups of small 2-rank

The simple groups of low 2-rank are mostly groups of Lie type of small rank over fields of odd characteristic, together with five alternating and seven characteristic 2 type and nine sporadic groups.

The simple groups of small 2-rank include:

Groups of 2-rank 0, in other words groups of odd order, which are all solvable by the Feit–Thompson theorem.
Groups of 2-rank 1. The Sylow 2-subgroups are either cyclic, which is easy to handle using the transfer map, or generalized quaternion, which are handled with the Brauer–Suzuki theorem: in particular there are no simple groups of 2-rank 1.
Groups of 2-rank 2. Alperin showed that the Sylow subgroup must be dihedral, quasidihedral, wreathed, or a Sylow 2-subgroup of U3(4). The first case was done by the Gorenstein–Walter theorem which showed that the only simple groups are isomorphic to L2(q) for q odd or A7, the second and third cases were done by the Alperin–Brauer–Gorenstein theorem which implies that the only simple groups are isomorphic to L3(q) or U3(q) for q odd or M11, and the last case was done by Lyons who showed that U3(4) is the only simple possibility.
Groups of sectional 2-rank at most 4, classified by the Gorenstein–Harada theorem.

The classification of groups of small 2-rank, especially ranks at most 2, makes heavy use of ordinary and modular character theory, which is almost never directly used elsewhere in the classification.

All groups not of small 2 rank can be split into two major classes: groups of component type and groups of characteristic 2 type. This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type. (For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.)
Groups of component type

A group is said to be of component type if for some centralizer C of an involution, C/O(C) has a component (where O(C) is the core of C, the maximal normal subgroup of odd order). These are more or less the groups of Lie type of odd characteristic of large rank, and alternating groups, together with some sporadic groups. A major step in this case is to eliminate the obstruction of the core of an involution. This is accomplished by the B-theorem, which states that every component of C/O(C) is the image of a component of C.

The idea is that these groups have a centralizer of an involution with a component that is a smaller quasisimple group, which can be assumed to be already known by induction. So to classify these groups one takes every central extension of every known finite simple group, and finds all simple groups with a centralizer of involution with this as a component. This gives a rather large number of different cases to check: there are not only 26 sporadic groups and 16 families of groups of Lie type and the alternating groups, but also many of the groups of small rank or over small fields behave differently from the general case and have to be treated separately, and the groups of Lie type of even and odd characteristic are also quite different.
Groups of characteristic 2 type

A group is of characteristic 2 type if the generalized Fitting subgroup F*(Y) of every 2-local subgroup Y is a 2-group. As the name suggests these are roughly the groups of Lie type over fields of characteristic 2, plus a handful of others that are alternating or sporadic or of odd characteristic. Their classification is divided into the small and large rank cases, where the rank is the largest rank of an odd abelian subgroup normalizing a nontrivial 2-subgroup, which is often (but not always) the same as the rank of a Cartan subalgebra when the group is a group of Lie type in characteristic 2.

The rank 1 groups are the thin groups, classified by Aschbacher, and the rank 2 ones are the notorious quasithin groups, classified by Aschbacher and Smith. These correspond roughly to groups of Lie type of ranks 1 or 2 over fields of characteristic 2.

Groups of rank at least 3 are further subdivided into 3 classes by the trichotomy theorem, proved by Aschbacher for rank 3 and by Gorenstein and Lyons for rank at least 4. The three classes are groups of GF(2) type (classified mainly by Timmesfeld), groups of "standard type" for some odd prime (classified by the Gilman–Griess theorem and work by several others), and groups of uniqueness type, where a result of Aschbacher implies that there are no simple groups. The general higher rank case consists mostly of the groups of Lie type over fields of characteristic 2 of rank at least 3 or 4.
Existence and uniqueness of the simple groups

The main part of the classification produces a characterization of each simple group. It is then necessary to check that there exists a simple group for each characterization and that it is unique. This gives a large number of separate problems for example, the original proofs of existence and uniqueness of the monster group totaled about 200 pages, and the identification of the Ree groups by Thompson and Bombieri was one of the hardest parts of the classification. Many of the existence proofs and some of the uniqueness proofs for the sporadic groups originally used computer calculations, most of which have since been replaced by shorter hand proofs.
History of the proof
Gorenstein's program

In 1972 Gorenstein (1979, Appendix) announced a program for completing the classification of finite simple groups, consisting of the following 16 steps:

1. Groups of low 2-rank. This was essentially done by Gorenstein and Harada, who classified the groups with sectional 2-rank at most 4. Most of the cases of 2-rank at most 2 had been done by the time Gorenstein announced his program.
2. The semisimplicity of 2-layers. The problem is to prove that the 2-layer of the centralizer of an involution in a simple group is semisimple.
3. Standard form in odd characteristic. If a group has an involution with a 2-component that is a group of Lie type of odd characteristic, the goal is to show that it has a centralizer of involution in "standard form" meaning that a centralizer of involution has a component that is of Lie type in odd characteristic and also has a centralizer of 2-rank 1.
4. Classification of groups of odd type. The problem is to show that if a group has a centralizer of involution in "standard form" then it is a group of Lie type of odd characteristic. This was solved by Aschbacher's classical involution theorem.
5. Quasi-standard form
6. Central involutions
7. Classification of alternating groups.
8. Some sporadic groups
9. Thin groups. The simple thin finite groups, those with 2-local p-rank at most 1 for odd primes p, were classified by Aschbacher in 1978
10. Groups with a strongly p-embedded subgroup for p odd
11. The signalizer functor method for odd primes. The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors. This was solved by McBride in 1982.
12. Groups of characteristic p type. This is the problem of groups with a strongly p-embedded 2-local subgroup with p odd, which was handled by Aschbacher.
13. Quasithin groups. A quasithin group is one whose 2-local subgroups have p-rank at most 2 for all odd primes p, and the problem is to classify the simple ones of characteristic 2 type. This was completed by Aschbacher and Smith in 2004.
14. Groups of low 2-local 3-rank. This was essentially solved by Aschbacher's trichotomy theorem for groups with e(G)=3. The main change is that 2-local 3-rank is replaced by 2-local p-rank for odd primes.
15. Centralizers of 3-elements in standard form. This was essentially done by the Trichotomy theorem.
16. Classification of simple groups of characteristic 2 type. This was handled by the Gilman-Griess theorem, with 3-elements replaced by p-elements for odd primes.

Many of the items in the list below are taken from Solomon (2001). The date given is usually the publication date of the complete proof of a result, which is sometimes several years later than the proof or first announcement of the result, so some of the items appear in the "wrong" order.
Publication date

 832 Galois introduces normal subgroups and finds the simple groups An (n ≥ 5) and PSL2(Fp) (p ≥ 5) 1854 Cayley defines abstract groups 1861 Mathieu describes the first two Mathieu groups M11, M12, the first sporadic simple groups, and announces the existence of M24. 1870 Jordan lists some simple groups: the alternating and projective special linear ones, and emphasizes the importance of the simple groups. 1872 Sylow proves the Sylow theorems 1873 Mathieu introduces three more Mathieu groups M22, M23, M24. 1892 Otto Hölder proves that the order of any nonabelian finite simple group must be a product of at least four (not necessarily distinct) primes, and asks for a classification of finite simple groups. 1893 Cole classifies simple groups of order up to 660 1896 Frobenius and Burnside begin the study of character theory of finite groups. 1899 Burnside classifies the simple groups such that the centralizer of every involution is a non-trivial elementary abelian 2-group. 1901 Frobenius proves that a Frobenius group has a Frobenius kernel, so in particular is not simple. 1901 Dickson defines classical groups over arbitrary finite fields, and exceptional groups of type G2 over fields of odd characteristic. 1901 Dickson introduces the exceptional finite simple groups of type E6. 1904 Burnside uses character theory to prove Burnside's theorem that the order of any non-abelian finite simple group must be divisible by at least 3 distinct primes. 1905 Dickson introduces simple groups of type G2 over fields of even characteristic 1911 Burnside conjectures that every non-abelian finite simple group has even order 1928 Hall proves the existence of Hall subgroups of solvable groups 1933 Hall begins his study of p-groups 1935 Brauer begins the study of modular characters. 1936 Zassenhaus classifies finite sharply 3-transitive permutation groups 1938 Fitting introduces the Fitting subgroup and proves Fitting's theorem that for solvable groups the Fitting subgroup contains its centralizer. 1942 Brauer describes the modular characters of a group divisible by a prime to the first power. 1954 Brauer classifies simple groups with GL2(Fq) as the centralizer of an involution. 1955 The Brauer–Fowler theorem implies that the number of finite simple groups with given centralizer of involution is finite, suggesting an attack on the classification using centralizers of involutions. 1955 Chevalley introduces the Chevalley groups, in particular introducing exceptional simple groups of types F4, E7, and E8. 1956 Hall–Higman theorem 1957 Suzuki shows that all finite simple CA groups of odd order are cyclic. 1958 The Brauer–Suzuki–Wall theorem characterizes the projective special linear groups of rank 1, and classifies the simple CA groups. 1959 Steinberg introduces the Steinberg groups, giving some new finite simple groups, of types 3 D4 and 2 E6 (the latter were independently found at about the same time by Jacques Tits). 1959 The Brauer–Suzuki theorem about groups with generalized quaternion Sylow 2-subgroups shows in particular that none of them are simple. 1960 Thompson proves that a group with a fixed-point-free automorphism of prime order is nilpotent. 1960 Feit, Hall, and Thompson show that all finite simple CN groups of odd order are cyclic. 1960 Suzuki introduces the Suzuki groups, with types 2 B2. 1961 Ree introduces the Ree groups, with types 2 F4 and 2 G2. 1963 Feit and Thompson prove the odd order theorem. 1964 Tits introduces BN pairs for groups of Lie type and finds the Tits group 1965 The Gorenstein–Walter theorem classifies groups with a dihedral Sylow 2-subgroup. 1966 Glauberman proves the Z* theorem 1966 Janko introduces the Janko group J1, the first new sporadic group for about a century. 1968 Glauberman proves the ZJ theorem 1968 Higman and Sims introduce the Higman–Sims group 1968 Conway introduces the Conway groups 1969 Walter's theorem classifies groups with abelian Sylow 2-subgroups 1969 Introduction of the Suzuki sporadic group, the Janko group J2, the Janko group J3, the McLaughlin group, and the Held group. 1969 Gorenstein introduces signalizer functors based on Thompson's ideas. 1970 MacWilliams shows that the 2-groups with no normal abelian subgroup of rank 3 have sectional 2-rank at most 4. (The simple groups with Sylow subgroups satisfying the latter condition were later classified by Gorenstein and Harada.) 1970 Bender introduced the generalized Fitting subgroup 1970 The Alperin–Brauer–Gorenstein theorem classifies groups with quasi-dihedral or wreathed Sylow 2-subgroups, completing the classification of the simple groups of 2-rank at most 2 1971 Fischer introduces the three Fischer groups 1971 Thompson classifies quadratic pairs 1971 Bender classifies group with a strongly embedded subgroup 1972 Gorenstein proposes a 16-step program for classifying finite simple groups the final classification follows his outline quite closely. 1972 Lyons introduces the Lyons group 1973 Rudvalis introduces the Rudvalis group 1973 Fischer discovers the baby monster group (unpublished), which Fischer and Griess use to discover the monster group, which in turn leads Thompson to the Thompson sporadic group and Norton to the Harada–Norton group (also found in a different way by Harada). 1974 Thompson classifies N-groups, groups all of whose local subgroups are solvable. 1974 The Gorenstein–Harada theorem classifies the simple groups of sectional 2-rank at most 4, dividing the remaining finite simple groups into those of component type and those of characteristic 2 type. 1974 Tits shows that groups with BN pairs of rank at least 3 are groups of Lie type 1974 Aschbacher classifies the groups with a proper 2-generated core 1975 Gorenstein and Walter prove the L-balance theorem 1976 Glauberman proves the solvable signalizer functor theorem 1976 Aschbacher proves the component theorem, showing roughly that groups of odd type satisfying some conditions have a component in standard form. The groups with a component of standard form were classified in a large collection of papers by many authors. 1976 O'Nan introduces the O'Nan group 1976 Janko introduces the Janko group J4, the last sporadic group to be discovered 1977 Aschbacher characterizes the groups of Lie type of odd characteristic in his classical involution theorem. After this theorem, which in some sense deals with "most" of the simple groups, it was generally felt that the end of the classification was in sight. 1978 Timmesfeld proves the O2 extraspecial theorem, breaking the classification of groups of GF(2)-type into several smaller problems. 1978 Aschbacher classifies the thin finite groups, which are mostly rank 1 groups of Lie type over fields of even characteristic. 1981 Bombieri uses elimination theory to complete Thompson's work on the characterization of Ree groups, one of the hardest steps of the classification. 1982 McBride proves the signalizer functor theorem for all finite groups. 1982 Griess constructs the monster group by hand 1983 The Gilman–Griess theorem classifies groups of characteristic 2 type and rank at least 4 with standard components, one of the three cases of the trichotomy theorem. 1983 Aschbacher proves that no finite group satisfies the hypothesis of the uniqueness case, one of the three cases given by the trichotomy theorem for groups of characteristic 2 type. 1983 Gorenstein and Lyons prove the trichotomy theorem for groups of characteristic 2 type and rank at least 4, while Aschbacher does the case of rank 3. This divides these groups into 3 subcases: the uniqueness case, groups of GF(2) type, and groups with a standard component. 1983 Gorenstein announces the proof of the classification is complete, somewhat prematurely as the proof of the quasithin case was incomplete. 1994 Gorenstein, Lyons, and Solomon begin publication of the revised classification 2004 Aschbacher and Smith publish their work on quasithin groups (which are mostly groups of Lie type of rank at most 2 over fields of even characteristic), filling the last gap in the classification known at that time. 2008 Harada and Solomon fill a minor gap in the classification by describing groups with a standard component that is a cover of the Mathieu group M22, a case that was accidentally omitted from the proof of the classification due to an error in the calculation of the Schur multiplier of M22. 2012 Georges Gonthier and collaborators announce a computer-checked version of the Feit-Thompson theorem using the Coq proof assistant. [1]

The proof of the theorem, as it stood around 1985 or so, can be called first generation. Because of the extreme length of the first generation proof, much effort has been devoted to finding a simpler proof, called a second-generation classification proof. This effort, called "revisionism", was originally led by Daniel Gorenstein.

As of 2005, six volumes of the second generation proof have been published (Gorenstein, Lyons & Solomon 1994, 1996, 1998, 1999, 2002, 2005). In 2012 Solomon estimated that the project would need another 5 volumes, but said that progress on them was slow. It is estimated that the new proof will eventually fill approximately 5,000 pages. (This length stems in part from second generation proof being written in a more relaxed style.) Aschbacher and Smith wrote their two volumes devoted to the quasithin case in such a way that those volumes can be part of the second generation proof.

Gorenstein and his collaborators have given several reasons why a simpler proof is possible.

• The most important is that the correct, final statement of the theorem is now known. Simpler techniques can be applied that are known to be adequate for the types of groups we know to be finite simple. In contrast, those who worked on the first generation proof did not know how many sporadic groups there were, and in fact some of the sporadic groups (e.g., the Janko groups) were discovered while proving other cases of the classification theorem. As a result, many of the pieces of the theorem were proved using techniques that were overly general.
• Because the conclusion was unknown, the first generation proof consists of many stand-alone theorems, dealing with important special cases. Much of the work of proving these theorems was devoted to the analysis of numerous special cases. Given a larger, orchestrated proof, dealing with many of these special cases can be postponed until the most powerful assumptions can be applied. The price paid under this revised strategy is that these first generation theorems no longer have comparatively short proofs, but instead rely on the complete classification.
• Many first generation theorems overlap, and so divide the possible cases in inefficient ways. As a result, families and subfamiles of finite simple groups were identified multiple times. The revised proof eliminates these redundancies by relying on a different subdivision of cases.
• Finite group theorists have more experience at this sort of exercise, and have new techniques at their disposal.

Aschbacher (2004) has called the work on the classification problem by Ulrich Meierfrankenfeld, Bernd Stellmacher, Gernot Stroth, and a few others, a third generation program. One goal of this is to treat all groups in characteristic 2 uniformly using the amalgam method.

Gorenstein has discussed some of the reasons why there might not be a short proof of the classification similar to the classification of compact Lie groups.

• The most obvious reason is that the list of simple groups is quite complicated: with 26 sporadic groups there are likely to be many special cases that have to be considered in any proof. So far no one has yet found a clean uniform description of the finite simple groups similar to the parameterization of the compact Lie groups by Dynkin diagrams.
• Atiyah and others have suggested that the classification ought to be simplified by constructing some geometric object that the groups act on and then classifying these geometric structures. The problem is that no-one has been able to suggest an easy way to find such a geometric structure associated to a simple group. In some sense the classification does work by finding geometric structures such as BN-pairs, but this only comes at the end of a very long and difficult analysis of the structure of a finite simple group.
• Another suggestion for simplifying the proof is to make greater use of representation theory. The problem here is that representation theory seems to require very tight control over the subgroups of a group in order to work well. For groups of small rank one has such control and representation theory works very well, but for groups of larger rank no-one has succeeded in using it to simplify the classification. In the early days of the classification there was considerable effort made to use representation theory, but this never achieved much success in the higher rank case.

Consequences of the classification

This section lists some results that have been proved using the classification of finite simple groups.

## 3 Answers 3

Recall that a quotient of an abelian group is again abelian. Notice that the subgroup generated by $leftlangle (0,1) ight angle$ in $mathbb_2 imes mathbb_4$ has $4$ elements, thus $mathbb_2 imes mathbb_4/leftlangle (0,1) ight angle$ has $2$ elements. By the classification of finite abelian groups we must have that this group is isomorphic to $mathbb_2$.

You can proceed in this way by doing the same for the other groups. If necessary you can look at order of elements to exclude certain possibilities.

Alternatively you can use the first isomorphism theorem:

If you see the proper morphism you can get the desired isomorphism immediately, but of course this method only works if you already know which group it should be.

## Mathematicians

Here are some of the mathematicians involved in my book Symmetry and the Monster.

### Late 18th century to mid 20th century

#### Joseph Louis Lagrange (1736–1813)

Born Guiseppe Lodovico Langrangia in northern Italy, he became professor in Berlin for more than 20 years, before taking up a position in Paris. He was one of the great mathematicians, working on many different aspects of mathematics: the three body problem differential equations number theory probability mechanics and the stability of the solar system. In particular he published an influential paper (Reflections on the Algebraic Solution of Equations) in 1770. This paper inspired the work of many others, including Galois. For biographical information see the St Andrews website, and Wikipedia.

#### Évariste Galois (1811–32)

Galois died in 1832 at the age of twenty. He was fatally wounded in a duel, but the night before the duel he wrote a long letter explaining his mathematical ideas. Among other things he studied the question of when an algebraic equation has solutions that can be expressed in terms of radicals (meaning square roots, cube roots, and so on). His method involved treating the solutions as objects that could be permuted among one another. The group of allowable permutations — the Galois group of the equation — reveals immediately whether the solutions can be expressed in terms of radicals, without knowing a single solution. Galois’ ideas were published in 1846, and have been extremely influential, leading to what is now known as Galois theory. For biographical information see the St Andrews website, and Wikipedia.

#### Augustin-Louis Cauchy (1789–1857)

Cauchy used clear and rigorous methods in studying calculus, and wrote several influential books on the topic. He also had wide-ranging interests and played a role in the early history of group theory. He proved a theorem showing that if the size of a group is divisible by a prime number p, then it has a subgroup of size p. For biographical information see the St Andrews website, and Wikipedia.

#### Camille Jordan (1838–1922)

Jordan’s 1870 Treatise on permutations and algebraic equations clarified and expanded the new subject of group theory, particularly in connection with Galois’s work. For biographical information see the St Andrews website, and Wikipedia.

#### Sophus Lie (1842–1899)

Lie was born in Oslo in 1842 (though at that time the city was called Christiania). He was a larger than life character who developed new methods for studying solutions to differential equations (equations involving rates of change). In this context he introduced groups in which each operation could be gradually modified — they are now known as Lie groups. Lie took up a chair in Germany in 1886, but returned to a chair in Norway a few months before his death in February 1899. For biographical information see the St Andrews website, and Wikipedia.

#### Wilhelm Killing (1847–1923)

Killing discovered Lie algebras independently of Lie’s work. He then went on to classify them, and from this classification the table of most finite ’symmetry atoms’ was created. For biographical information see the St Andrews website, and Wikipedia.

#### Élie Cartan (1869–1951)

In his PhD thesis, Cartan revised Killing’s proofs of the classification of Lie algebras. He then went on to make significant contributions to differential equations and geometry. More details, click here. For biographical information see the St Andrews website, and Wikipedia.

#### William Burnside (1852–1927)

Burnside wrote the first book on group theory in English, published in 1897, and developed the subject from the modern abstract point of view. In 1904 he proved that the size of any finite simple group that is non-cyclic must be divisible by at least three different prime numbers. For biographical information see the St Andrews website, and Wikipedia.

#### Leonard Eugene Dickson (1874–1954)

In 1901 he published a book showing how to obtain finite versions for most families of Lie groups. This was the start of the ‘periodic table’ of finite simple groups. For biographical information see the St Andrews website, and Wikipedia.

#### Richard Brauer (1901–77)

Brauer founded the ‘cross-section’ (i.e. involution centralizer) approach to classifying the finite simple groups. He also did leading work on the character theory of finite groups. For biographical information see the St Andrews website, and Wikipedia.

#### Claude Chevalley (1909–84)

Chevalley worked on group theory and ring theory and in 1955 published a paper showing how to obtain finite versions of Lie groups in all families. For biographical information see the St Andrews website, and Wikipedia.

#### Jacques Tits (1930–

Like Chevalley, Tits was also pursuing finite versions of Lie groups in all families, but in a geometric way rather than using Chevalley’s algebraic approach. It led him to create the theory of buildings (which are ‘multi-crystals’, not buildings in the usual sense), which he went on to develop in other important ways. In 2008, Tits was awarded the Abel Prize, jointly with John Thompson. For biographical information, see the St Andrews website, and Wikipedia.

#### Walter Feit (1930–2004)

Feit was an expert on the character theory of finite groups, and collaborated with John Thompson to prove the celebrated theorem (the Feit-Thompson theorem) showing that a finite simple group that is not cyclic must have even size. For biographical information, see the St Andrews website, and Wikipedia.

#### John Thompson (1932–

Thompson’s early work led to his collaboration with Walter Feit on the great Feit-Thompson theorem (above). He went on to deal with the cross-section method of classifying finite simple groups, and was involved in studying the Monster and the new simple groups inside it, one of which is named after him. In 2008, Thompson was awarded the Abel Prize, jointly with Jacques Tits. For biographical information see the St Andrews website, and Wikipedia.

#### Daniel Gorenstein (1923–1992)

Gorenstein was the first person to put forward a plan for classifying all the finite simple groups, and he was closely involved with steering this project forward. When it appeared complete, he started the project, in collaboration with Lyons and Solomon, of revising and rewriting it so that it would stand the scrutiny of future generations. For biographical information see the St Andrews website, and Wikipedia.

### The Classification and Discovery of the Sporadic Groups

A great many mathematicians were involved in the Classification project, but only a few are mentioned in the book, and the same is true here. No disrespect is intended to those who are missing—only people whose work appears in the book are mentioned here, and the book is not a complete history of the Classification. For that one needs to read the books by Gorenstein, and by Gorenstein, Lyons and Solomon. Here the main topic is the discovery of the sporadic groups.

#### Émile Mathieu (1835–90)

A French mathematical physicist who, as a student, studied permutation groups that are multiply transitive. His results yielded five simple groups that are not of ‘Lie type’. These are the Mathieu groups M11, M12, M22, M23 and M24. For biographical information see the St Andrews website, and Wikipedia.

#### Ernst Witt (1911–91)

Witt created the Witt design on 24 symbols. It gives a simple way of understanding the Mathieu groups, and proves their existence. For biographical information see the St Andrews website, and Wikipedia.

#### John Leech (1926–92)

Leech discovered the Leech Lattice in 24 dimensions by using the Witt design, and started studying its symmetry group. For biographical information see the St Andrews website, and Wikipedia.

#### John Conway (1937–

Conway studied the symmetries of the Leech Lattice from which he produced three new finite simple groups, along with others that had already been found by other methods. He later worked on the Monster and its moonshine connections. For biographical information see the St Andrews website, and Wikipedia.

#### Zvonimir Janko (1932–

In 1966, Janko published the first new exception since Mathieu’s groups a century earlier. It is now known as J1. He went on to discover three more: J2, J3 and J4. All Janko’s sporadic groups were discovered by the cross-section (involution centralizer) method. For more information see Wikipedia.

#### Michio Suzuki (1926–98)

Suzuki made important contributions to the classification project in the early days and proved a version of the Feit-Thompson theorem in an important special case. In the early 1960s he also discovered a new family of finite simple groups that subsequently turned out to be groups of Lie type. Later in the 1960s he discovered a sporadic group that bears his name. For biographical information, see Wikipedia, and the website from the University of Illinois where he worked.

#### Bernd Fischer (1936–

Fischer discovered several sporadic groups, three of which are known by his name. These are the Fischer groups Fi22, Fi23 and Fi24 (the last one is not simple but contains a large simple subgroup). Fischer also discovered the Baby Monster, from which emerged the Monster. This in turn produced two new sporadic groups, which are named after those who did most of the work on them: Thompson in one case, and Harada and Norton in the other. For further information see Wikipedia.

#### Donald Livingstone (1924–2001)

Livingstone and Fischer together created the character table of the Monster, with Michael Thorne writing the computer programs that they needed for the calculations.

#### Marshall Hall (1910–90)

Hall constructed Janko’s group J2 as a group of permutations on 100 symbols. For biographical information see the St Andrews website, and Wikipedia.

#### Graham Higman (1917–2008)

Higman worked on the construction of several sporadic groups that had been discovered by the cross-section (involution centralizer) method. For biographical information see the St Andrews website, or Wikipedia.

#### Donald Higman (1928–2006)

Higman, in collaboration with Charles Sims, adapted Hall’s construction of J2 to produce another group of permutations on 100 symbols that was a new finite simple group. This is the Higman-Sims group. For more information see Wikipedia.

#### Charles Sims (1938–

In addition to being a co-discoverer of the Higman-Sims group, Sims used permutation techniques to construct several other sporadic groups. In collaboration with Jeffrey Leon he constructed the Baby Monster. For biographical information see Wikipedia.

#### Robert Griess (1945–

Griess predicted the Monster independently of Fischer, using Fischer’s Baby Monster. He later constructed the Monster as the group of symmetries for an algebra in 196,884 dimensions. For further information see his homepage, or Wikipedia.

#### John McLaughlin (1923–2001)

McLaughlin created a new sporadic group (the McLaughlin group) as a group of permutations.

#### Arunas Rudvalis (1945–

Rudvalis predicted the existence of a new sporadic group (the Rudvalis group) as a group of permutations. It was later constructed by Conway and David Wales. For more information see Wikipedia.

#### Dieter Held (1936–

Held discovered the sporadic group that bears his name. He used the cross-section (involution centralizer) method, and the group was later constructed by Graham Higman and John McKay. For more information see Wikipedia.

#### Michael O’Nan (1943–

O’Nan discovered the sporadic group that bears his name. He used the cross-section (involution centralizer) method, and the group was later constructed by Charles Sims.

#### Richard Lyons (1945–

Lyons discovered the sporadic group that bears his name. He used the cross-section (involution centralizer) method, and the group was later constructed by Charles Sims. Then with Ronald Solomon and Daniel Gorenstein he undertook the Revision of the Classification, a project that continues to this day. For more information see Wikipedia.

#### Koichiro Harada (1941–

Harada studied one of the two previously undiscovered simple groups that emerged as subgroups of the Monster. It is named after him and Simon Norton. For more information see Wikipedia.

#### Simon Norton (1952–

Norton calculated a large amount of information on the Harada-Norton group, and on the Monster itself. He also collaborated with Conway on the strange moonshine connections with the j‑function in number theory. For more information see Wikipedia.

#### John McKay (1939–

McKay made the first observation of a numerical coincidence between the Monster and the j‑function in number theory. He also made other intriguing observations, some of which have since been elucidated. For more information see Wikipedia.

#### Igor Frenkel (1952–

With James Lepowsky and Arne Meurman, he created the Moonshine module, connecting the Monster and the j‑function. For more information, see Wikipedia.

#### James Lepowsky (1944–

With James Lepowsky and Arne Meurman, he created the Moonshine module, connecting the Monster and the j‑function. For more information see Wikipedia.

#### Arne Meurman (1956–

With James Lepowsky and Arne Meurman, he created the Moonshine module, connecting the Monster and the j‑function. For more information see Wikipedia.

#### Richard Borcherds (1959–

Borcherds created a Monster Lie algebra that led him to a proof of the Conway-Norton conjectures for the Moonshine module, an achievement for which he was awarded the Fields Medal in 1998. For biographical information see the St Andrews website, and Wikipedia.

#### Michael Aschbacher (1944–

Aschbacher was the greatest contributor to the Classification program, apart from Thompson. He and Stephen Smith eventually filled in the missing part of the program, the quasi-thin case. For some biographical information, see Wikipedia.

#### Stephen Smith (1948–

In joint work, Smith and Aschbacher filled in a gap in the Classification by finally nailing the quasi-thin case. For biographical information see his homepage.

#### Ronald Solomon (1948–

Solomon, together with Richard Lyons and Daniel Gorenstein undertook the Revision of the Classification, a project that continues to this day. For biographical information see Wikipedia.

## References

B. Alspach, M. Conder, D. Marušič, and M.Y. Xu, “A classification of 2-arc-transitive circulants,” J. Algebraic Combin. 5 (1996), 83–86.

C.Y. Chao, “On the classification of symmetric graphs with a prime number of vertices,” Trans. Amer. Math. Soc. 158 (1971), 247–256.

C.Y. Chao and J.G. Wells, “A class of vertex-transitive digraphs,” J. Combin. Theory Ser. B 14 (1973), 246–255.

S.A. Evdokimov and I.N. Ponomarenko, “Characterization of cyclotomic schemes and normal Schur rings over a cyclic group, (Russian),” Algebra i Analiz 14(2) (2002), 11–55.

W. Feit, “Some consequences of the classification of finite simple groups,” The Santa Cruz Conference on Finite Groups Univ. California, Santa Cruz, Calif., 1979, pp. 175–181, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980.

C.D. Godsil, “On the full automorphism group of a graph,” Combinatorica 1 (1981), 243–256.

D. Gorenstein, Finite Simple Groups, Plenum Press, New York, 1982.

G. Jones, “Cyclic regular subgroups of primitive permutation groups,” J. Group Theory 5(4) (2002), 403–407.

W. Kantor, Some consequences of the classification of finite simple groups, in Finite Groups—Coming of Age, John McKay (Ed.), Amer. Math. Soc., 1982.

I. Kovács, “Classifying arc-transitive circulants,” J. Algebraic Combin. 20 (2004), 353–358.

K.H. Leung and S.H. Man, “On Schur rings over cyclic groups,” II, J. Algebra 183 (1996), 173–285.

K.H. Leung and S. H. Man, “On Schur rings over cyclic groups,” Israel J. Math. 106 (1998), 251–267.

C.H. Li, “The finite primitive permutation groups containing an abelian regular subgroup,” Proc. London Math. Soc. 87 (2003), 725–748.

C.H. Li, “Edge-transitive Cayley graphs and rotary Cayley maps,” Trans. Amer. Math. Soc. (to appear)

C.H. Li, D. Marušič, and J. Morris, “A classification of circulant arc-transitive graphs of square-free order,” J. Algebraic Combin. 14 (2001), 145–151.

J.X. Meng and J.Z. Wang, “A classification of 2-arc-transitive circulant digraphs,” Disc. Math. 222 (2000), 281–284.

H. Wielandt, Finite Permutation Groups, Academic Press, New York, 1964.

H. Wielandt, Permutation groups through invariant relations and invariant functions, in Mathematische Werke/Mathematical works. Vol. 1. Group theory, Bertram Huppert and Hans Schneider (Eds.), Walter de Gruyter Co., Berlin, 1994, pp. 237–266.

## Everything about Group Theory

This recurring thread will be a place to ask questions and discuss famous/well-known/surprising results, clever and elegant proofs, or interesting open problems related to the topic of the week. Experts in the topic are especially encouraged to contribute and participate in these threads.

Today's topic is Group Theory. Next week's topic will be Number Theory. Next-next week's topic will be Analysis of PDEs.

So, I'm this guy. I've written a lot of stuff about group theory on the Internet, the coolest of which are (if you'll excuse the plug):

A list of my MSE posts about group theory, including a couple open questions, my big list of finite groups, and a few other neat posts.

My upcoming paper linking solvable group structure to graph theory.

Iɽ be happy to answer any group theory questions people have, or just hang out in this thread and chat a bit. Hi guys.

Following Alexander Gruber's (IAmVeryStupid's) lead, I am this guy. I am a professional group theorist specializing in geometric group theory and its connections with dynamical systems and fractal geometry. I have the following background:

I have four published papers (1, 2, 3, 4) and two recent preprints (5, 6) on group theory.

I've taught undergraduate abstract algebra three times at Bard college.

I have written several MSE posts on group theory, and I have also contributed to a few Wikipedia articles.

I wrote this reddit post describing the state of modern group theory as I see it.

Hi everyone! Iɽ be happy to answer questions or just chat for a while.

What's your favourite group?

Ah, I have seen your posts a lot!

In particular I fondly recall this thread and your fantastic answer.

Whilst you are here, why don't you give us your favourite finite group?

I had a look at your preprint (arXiv link), and I'm curious whether anything is known about the prime graphs of infinite solvable groups.

Thanks for your contribution! I'm n00b at group theory, but I'm in a line of work where knowing more would definitely be useful!

A couple questions, ELI 1st/2nd year college undergrad please:

What's an Abelian group/special unitary group?

How are different group defined. What is isomorphism?

What are 'interesting' groups, as far as mathematicians/physicist are concerned.

Thanks so much! If you've answered some of these in the past, a link is fine as well.

As briefly as possible, explain what group theory is, and what you do with it.

Hello there! I'm an undergraduate senior at UF majoring in Mathematics and Philosophy, about to start a PhD program in civil engineering here as well. I think it's funny that I've never seen you around.

How would you go about understanding a specific group?

What Im looking for is a collection of clues you could look for. Something like: "If the center is large then the group might be . " "If the automorpism group is abelian then you might want to look at. "

So Im not looking so much for a list of theorems as I am looking for a systematic approach to study a specifik group.

What would you say are the results that one needs to know to consider before he can consider himself a finite group theorist? Feit-Thompson? Stuff about modular representations of finite groups? The classification? Waring problem for finite groups?

What are the books that you would recommend to someone who wanted to go from being a complete beginner to an expert on the subject of finite groups?

One thing I am compelled to write about are Sylow's Theorems, which are an incredibly powerful tool for classifying finite groups.

If G is a finite group of order m*p l (where p does not divide m). then a Sylow p subgroup is a subgroup of order p l .

Sylow's three theorems are:

For all prime factors p of the order of the group, there exists a Sylow p subgroup.

For all prime factors p, all Sylow p subgroups are conjugate.

For a prime factor p, there are exactly N Sylow p subgroups, where N divides m, and N = 1 mod p.

The proof of these 3 theorems were the bane of my existence for a whole semester, beautiful though they are.

I have to say, I've never been very fond of the Sylow theorems. They are often covered in introductory abstract algebra courses, but I think this reflects an old-fashioned view towards group theory, where the core of the subject was the classification of finite simple groups.

Mathematics has moved on since then, and the Sylow theorems have become less and less relevant. I don't think they deserve to be covered in a typical undergraduate abstract algebra class, and I'm not even sure that they ought to be covered in a typical graduate algebra class. In my mind, it would make more sense to talk more about matrix groups and representations, or to discuss some basic facts about infinite groups, e.g. classifying subgroups of free groups.

This is not to say that I don't appreciate the Sylow theorems aesthetically. It's just that pedagogically I think they are vastly overemphasized.

## 5.4: Classifying Finite Groups - Mathematics

Note! The statement in 9(b) is false as written. You should prove the given statement under the assumption that I is generated by a polynomial f that does not have repeated factors in its factorization.

Summary Groups, rings and field. More detail: Groups acting on sets, examples of finite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals polynomial rings over a field PID and non-PID. Unique factorization domains.
This is a "Writing in the major" (WIM) class.

Assessment: Combination of weekly homework (25%), WIM assignment (15%), midterm (25%), and final (35%). There will be weekly homework assignments.

Text: The course text will be Algebra by Dummit and Foote. We will cover roughly the first 8 chapters.

Office hours: My office hours will be MWF after class (9:50--10:50). The CA is Jeremy Leach. His office hours: Tuesday 3-5, WF 12:15--2:15.

WIM assignment info: Draft due May 16, final version due May 27.

The WIM Assignment is to write a short (around 5 pages) exposition of the classification theoreom for finite abelian groups. (If you want, you can do another topic of your choice, but if so you should discuss it with me to make sure you know what you're getting into.) You can find a statement of a more general theorem in 5.2 and a proof in 6.1. You'll hand in a draft by May 16 to Jeremy Leach he will get you back comments, and you should hand in the final version by Friday, May 27. You are very welcome to talk to either Jeremy or me about this before the draft to get even earlier feedback you are also welcome (and encouraged) to hand in the draft earlier. Details. Homework sets will be posted here. Here u.x.y = problem y from section u.x of Dummitt and Foote.
Homework 1, due Friday April 8. 1.1.24, 1.1.25, 1.1.31, 1.2.13, 1.3.2, 1.3.18, 1.5.1, 1.6.5, 1.6.6, 1.6.17, 1.6.24. (You may need to read some of sections 1.1--1.6).
Solutions by Jeremy Leach.

Homework 2, due Friday April 15: 1.7.8, 1.7.16, 1.7.19, 1.7.23, 2.1.12, 2.1.14, 2.1.5, 2.2.4, 2.2.5, 2.2.12, 2.3.10, 2.3.16, 2.3.21, 2.3.25. Solutions.

Homework 3, posted Monday April 18, due Monday April 25: 2.4.2, 2.4.7, 2.4.8, 2.5.2, 2.5.4, 3.1.1, 3.1.3, 3.1.14, 3.1.32, 3.1.34, 3.1.42. Bonus question: explain why, if A and B are two rotations in three dimensions, the products AB and BA are rotations by the same angle, although possibly through different axes. Solutions.

Homework 4, posted Sunday April 24, not for assessment because of the midterm, but I recommend you try it anyway: 3.2.5, 3.2.7, 3.2.8, 3.2.11, 3.2.12, 3.3.7, 3.3.8, 3.5.4, 3.5.9, 3.5. 11. Bonus question: If G is a group and H a subgroup of index 5, prove that there is a normal subgroup N of G of index at most 120, contained in H.

Homework 5, due Friday May 6: 3.5.5, 3.5.12, 4.1.1, 4.1.6, 4.1.7, 4.1.8, 4.1.10, 4.2.2, 4.2.4, 4.2.7 4.2.11, 4.2.12. Bonus question: Discuss the solvability of the 15 puzzle.

Homework 6, due Friday May 13: 4.3.3, 4.3.5, 4.3.9, 4.3.11, 4.3.13, 4.3.22, 4.3.30, 4.3.36, 4.4.2, 4.4.18.

Homework 7, due Friday May 20: 4.5.4, 4.5.6, 4.5.15, 4.5.18, 4.5.33, 4.6.1, 4.6.2, 4.6.4.
Bonus question: Let f: G-->G' be a surjective homomorphism with kernel K. We assume G is finite. Let P, Q be Sylow p-subgroups of G. Show that f(P) is a p-Sylow subgroup of G'. Show that f(P)= f(Q) if and only if P and Q are conjugate by an element of K. solutions.

Homework 8, due Friday May 27: 7.1.1, 7.1.5, 7.1.11, 7.1.12, 7.1.13, 7.2.1, 7.3.1, 7.3.2, 7.3.6, 7.3.10, 7.4.8, 7.4.15, 7.6.3. solutions.

"Homework 9": (not for assessment, just some more practice problems related to the material in the last week of course): 8.1.7, 8.2.5, 8.3.3, 8.3.6, 8.3.7, 8.3.8.

## 2 Answers 2

Finite topologies and finite preorders (reflexive & transitive relations) are equivalent:

Let $T$ be a topological space with finite topology $mathcal$. Define $leq$ on $T$ by: $xleq y Leftrightarrow forall Uin mathcal : xin U Rightarrow yin U$

Then $leq$ is clearly a preorder, called the specialization order of $T$.

Given a preorder $leq$ on $T$, define the set $mathcal$ to be set of all upwards-closed sets in $(T,leq)$, that is all sets $U$ with:

$forall x,yin T : xleq y ext < and >xin U Rightarrow yin U$

Then $mathcal$ is a topology, called the specialization topology or Alexandroff topology of $(T,leq)$.

The constructions are functorial and can be turned into an equivalence of categories $mathsf$ and $mathsf$ (I don't have time to work out the details right now, however).

## About the Author

Brian S. Everitt, Head of the Biostatistics and Computing Department and Professor of Behavioural Statistics, Kings College London. He has authored/ co-authored over 50 books on statistics and approximately 100 papers and other articles, and is also joint editor of Statistical Methods in Medical Research.

Dr Sabine Landau, Head of Department of Biostatistics, Institute of Psychiatry, Kings College London.

Dr Morven Leese, Health Service and Population Research, Institute of Psychiatry, Kings College London.

Dr Daniel Stahl, Deptartment of Biostatistics & Computing, Institute of Psychiatry, Kings College London.