Finite nonabelian simple group

A finite nonabelian simple group is a finite group that is a nonabelian group and a simple group. Every such group is a perfect group because its commutator subgroup is a nontrivial normal subgroup.

Let $p$ be the smallest prime divisor of the order of a finite nonabelian simple group. A cyclic Sylow subgroup would give a normal p-complement, so if $p^3$ does not divide the order, the Sylow subgroup must be $C_p^2$. Conjugation embeds its normalizer modulo centralizer into $G{L}_2(p)$, and this quotient has order prime to $p$. For odd $p$, every other prime divisor of $|G{L}_2(p)|=p(p-1{)}^2(p+1)$ is smaller than $p$, again forcing trivial conjugation and a contradiction to the Burnside transfer theorem. For $p=2$, nontrivial conjugation must have order three, giving $12\mid |G|$. The groups $PS{L}_2(5^{2m+1})$ give infinitely many examples with exactly two factors of two in their orders.

The natural actions of these five sporadic simple groups have degrees 11, 12, 22, 23 and 24, and respective transitivity degrees 4, 5, 3, 4 and 5. They arise from exceptional finite designs and codes.

These simple groups have order $q^3(q^3+1)(q-1)$ and a two-transitive action on $q^3+1$ points. The restriction excludes the smallest nonsimple parameter; this family differs from the Ree groups of type $F_4$.

These simple groups have order $q^2(q^2+1)(q-1)$ and a two-transitive action on a Suzuki ovoid of $q^2+1$ points. The point-stabilizer root group is regular on the remaining points.