= Least-prime divisibility constraint for a finite simple group
{title2=$p^3\mid |G|\quad\text{or}\quad p=2,\ 12\mid |G|$}
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 $GL_2(p)$, and this quotient has order prime to $p$. For odd $p$, every other prime divisor of $|GL_2(p)|=p(p-1)^2(p+1)$ 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> $PSL_2(5^{2m+1})$ give infinitely many examples with exactly two factors of two in their orders.
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