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| Classification of finite simple groups | 1/4 | https://en.wikipedia.org/wiki/Classification_of_finite_simple_groups | reference | science, encyclopedia | 2026-05-05T09:08:19.935542+00:00 | kb-cron |
In mathematics, the classification of finite simple groups (popularly called the enormous theorem) is a result of group theory stating that every finite simple group is either cyclic, or alternating, or belongs to a broad infinite class called the groups of Lie type, or else it is one of twenty-six exceptions, called sporadic (the Tits group is sometimes regarded as a sporadic group because it is not strictly a group of Lie type, in which case there would be 27 sporadic groups). The proof consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004. Simple groups can be seen as the basic building blocks of all finite groups, 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 from integer factorization is that such "building blocks" do not necessarily determine a unique 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. Daniel Gorenstein (1923–1992), Richard Lyons, and Ronald Solomon are gradually publishing a simplified and revised version of the proof.
== Statement of the classification theorem ==
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 except for the cyclic group of order two. 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.