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

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