Simple Groups and the Classification

Simple groups — those with no non-trivial proper normal subgroup — are the irreducible building blocks of finite group theory: by Jordan–Hölder every finite group is assembled from simple composition factors. The complete list of finite simple groups, one of the largest theorems in mathematics, is surveyed here. This page builds on series-solvable.md, sylow.md, and permutation-groups.md.

Definition and role

A group $G\ne \{e\}$ is simple if its only normal subgroups are $\{e\}$ and $G$. Simple groups cannot be broken down by quotienting, so they are the atoms of the composition series. Two flavours:

• Abelian simple groups are exactly the cyclic groups ${\mathbb{Z}}_p$ of prime order (any subgroup of an abelian group is normal, so simplicity forces no proper subgroups, i.e. prime order).
• Non-abelian simple groups are the interesting ones. The smallest is $A_5$ (order $60$); Sylow counting shows there are no non-abelian simple groups of order below $60$.

The classification (CFSG)

Classification of Finite Simple Groups. Every finite simple group is isomorphic to one of:

1. a cyclic group ${\mathbb{Z}}_p$ of prime order;
2. an alternating group $A_n$, $n\ge 5$;
3. a group of Lie type — finite analogues of the classical and exceptional Lie groups, built over finite fields ${\mathbb{F}}_q$ (e.g. $PSL(n,q)$, $PSp(2n,q)$, the Chevalley and twisted Steinberg/Suzuki/Ree families);
4. one of 26 sporadic groups belonging to none of the above families.

The proof runs to tens of thousands of journal pages assembled over decades (Gorenstein's program), with a "second-generation" streamlined proof still in progress. It is among the deepest results in mathematics, and it makes the composition factors of any finite group members of an explicit list.

The families

• Cyclic ${\mathbb{Z}}_p$ — the abelian atoms; infinitely many, one per prime.
• Alternating $A_n$ — simple for $n\ge 5$ (permutation-groups.md); the "permutation" family.
• Groups of Lie type — by far the largest source, roughly "matrix groups over ${\mathbb{F}}_q$ modulo their centre". Example: $PSL(2,7)\cong PSL(3,2)$ of order $168$, the second-smallest non-abelian simple group. These are the finite shadows of the Lie-group classification.

The sporadic groups

The 26 sporadic simple groups fit no infinite family. They include the five Mathieu groups $M_{11},M_{12},M_{22},M_{23},M_{24}$ (19th-century multiply-transitive permutation groups, tied to the exceptional $S_6$ automorphism of permutation-groups.md), the Leech lattice groups, and the Monster $\mathbb{M}$ — the largest, of order $$|\mathbb{M}|=808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000\approx 8\times 10^{53}.$$ Twenty of the sporadics are subquotients of the Monster (the "Happy Family"); the remaining six are "pariahs". The Monster's mysterious connection to modular forms — monstrous moonshine — links finite group theory to number theory and string theory.

Why the list matters

The classification reduces countless structural questions to checking a finite menu. Combined with Jordan–Hölder, it means the composition factors of every finite group are known explicitly; combined with the extension theory of how those factors can be glued, it is in principle a complete description of all finite groups. It is the finite counterpart of the Lie-algebra classification — both reduce "all objects" to "simple objects + assembly".

References

• Wilson, The Finite Simple Groups (Springer GTM 251).
• Aschbacher, Finite Group Theory.
• Ronan, Symmetry and the Monster — accessible history of the classification.