Simple groups of order $2^{a}3^{b}5^{c}7^{d}p$

Abstract:

Let ${\operatorname {PSL}}(n,q)$ denote the projective special linear group of degree $n$ over ${\text {GF}}(q)$, the field with $q$ elements. The following theorem is proved. Theorem. Let $G$ be a simple group of order ${2^a}{3^b}{5^c}{7^d}p,a > 0,p$ an odd prime. If the index of a Sylow $p$-subgroup of $G$ in its normalizer is two, then $G$ is isomorphic to one of the groups, ${\operatorname {PSL}}(2,5),{\operatorname {PSL}}(2,7),{\operatorname {PSL}}(2,9),{\operatorname {PSL}}((2,8),{\operatorname {PSL}}(2,16),{\operatorname {PSL}}(2,25),{\operatorname {PSL}}(2,27),{\operatorname {PSL}}(2,81)$, and ${\operatorname {PSL}}(3,4)$.