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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Author: Leo J. Alex
Journal: Trans. Amer. Math. Soc. 173 (1972), 389-399
MSC: Primary 20D05
DOI: https://doi.org/10.1090/S0002-9947-1972-0318291-1
MathSciNet review: 0318291
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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 ${\operatorname {PSL}}(3,4)$.


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Keywords: Finite simple group classification, class algebra coefficient, characters of finite groups, principal <IMG WIDTH="16" HEIGHT="37" ALIGN="MIDDLE" BORDER="0" SRC="images/img9.gif" ALT="$p$">-block
Article copyright: © Copyright 1972 American Mathematical Society