List of Finite Simple Groups

Introduction

This article lists the finite simple groups, the atomic objects of finite group theory, following the four families of the classification: the cyclic groups of prime order, the alternating groups of degree at least five, the groups of Lie type in their sixteen infinite families, and the twenty-six sporadic groups. For each family the list records the order, and for the sporadic groups it records the six subfamilies into which they fall.

Every entry points to the article that introduces the group and states the order recorded there. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being a small member of a family that fails simplicity, with the failure named and the article that records it.

The Statement of the Classification

A finite group is built from its simple composition factors by Jordan–Hölder, so the classification of the simple groups classifies the building blocks. The theorem states that every finite simple group is a cyclic group of prime order, an alternating group $A_n$ with $n \geq 5$, a group of Lie type, or one of the twenty-six sporadic groups. The smallest nonabelian finite simple group is $A_5$, of order $60$, and there is no nonabelian simple group of order less than $60$.

The Cyclic Groups of Prime Order

The cyclic groups $\mathbb{Z}/p\mathbb{Z}$ of prime order form the abelian family of the classification, and they are the only abelian finite simple groups. For odd $p$ the order is odd, so these are exactly the odd-order members of the list.

The Alternating Groups

The alternating group $A_n$ is the subgroup of even permutations of $S_n$, of order $n!/2$, and it is simple for every $n \geq 5$. The low-degree cases are the exceptions: $A_4$ of order $12$ has a normal Klein four-subgroup, and $A_3 \cong C_3$ and $A_2 = 1$ fall outside the defining range.

The Groups of Lie Type

The groups of Lie type are the finite groups obtained from a simple linear algebraic group over a finite field, as its $\mathbb{F}_q$-points modulo the centre. There are sixteen infinite families: the classical families $\mathrm{PSL}_n(q)$, $\mathrm{PSp}_{2n}(q)$, $\mathrm{PSU}_n(q)$ and $\mathrm{P}\Omega_n(q)$, the exceptional families $G_2(q)$, $F_4(q)$, $E_6(q)$, $E_7(q)$ and $E_8(q)$, the twisted families ${}^2B_2$, ${}^2G_2$, ${}^2F_4$ and ${}^3D_4$, with the Tits group ${}^2F_4(2)'$ simple but not itself of Lie type in the strict sense.

The Twenty-Six Sporadic Groups

The sporadic groups are the twenty-six finite simple groups outside the three infinite families. They fall into six subfamilies: the five Mathieu groups, the seven Conway and Leech-related groups, the three Fischer groups, the three Held, Thompson and Harada–Norton groups, the six pariahs, and the two large sporadics, the Baby Monster and the Monster.

Small Orders and Exceptional Isomorphisms

The four families intersect in a finite list of small cases, which the classification registers so that no simple group is counted twice. The coincidences among the simple groups include $A_5 \cong \mathrm{PSL}_2(4) \cong \mathrm{PSL}_2(5)$, $A_6 \cong \mathrm{PSL}_2(9)$, $A_8 \cong \mathrm{PSL}_4(2)$, $\mathrm{PSL}_2(7) \cong \mathrm{PSL}_3(2)$, $\mathrm{PSU}_4(2) \cong \mathrm{PSp}_4(3)$, $G_2(2)' \cong \mathrm{PSU}_3(3)$ and ${}^2G_2(3)' \cong \mathrm{PSL}_2(8)$. The smallest orders of nonabelian simple groups are $60$, $168$, $360$, $504$, $660$ and $1092$.

Summary

The list gathers the finite simple groups. The abelian family is the cyclic groups of prime order; the alternating family is the groups $A_n$ for $n \geq 5$, of order $n!/2$, beginning with $A_5$ of order $60$; the Lie family consists of the sixteen infinite families $\mathrm{PSL}_n(q)$, $\mathrm{PSp}_{2n}(q)$, $\mathrm{PSU}_n(q)$, $\mathrm{P}\Omega_n(q)$, $G_2(q)$, $F_4(q)$, $E_6(q)$, $E_7(q)$, $E_8(q)$, ${}^2B_2$, ${}^2G_2$, ${}^2F_4$ and ${}^3D_4$, together with the Tits group; and the sporadic family is the twenty-six groups, in the six subfamilies of Mathieu, Conway and Leech, Fischer, Held–Thompson–Harada–Norton, pariah and large sporadic. The non-examples — $A_4$, the cyclic group of composite order, $\mathrm{PSL}_2(2)$, $\mathrm{PSL}_2(3)$, $\mathrm{PSU}_3(2)$, $G_2(2)$, ${}^2G_2(3)$, ${}^2F_4(2)$, $\mathrm{Sp}_4(2)$ and the groups of two-prime order — are the small or degenerate members that fail simplicity.

Summary of Notation

The article denotes its objects by name; the symbols appearing in the tables are those of the introducing articles.

Further Reading

• Daniel Gorenstein, Richard Lyons and Ronald Solomon, The Classification of the Finite Simple Groups (American Mathematical Society, 1994–), for the statement, the four families and the second-generation proof.
• John Conway, Robert Curtis, Simon Norton, Richard Parker and Robert Wilson, Atlas of Finite Groups (Oxford University Press, 1985), for the orders, the subfamily grouping and the exceptional isomorphisms of the finite simple groups.
• Roger Carter, Simple Groups of Lie Type (Wiley, 1972), for the sixteen families of Lie type, their orders and their small degenerate members.
• Robert Wilson, The Finite Simple Groups (Springer, 2009), for a single-volume account of the four families with the orders tabulated.