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Solvable and Nilpotent Subgroups of GL(n,qm)

Published online by Cambridge University Press:  20 November 2018

Thomas R. Wolf
Thomas R. Wolf*
Affiliation:
University of Wisconsin-Milwaukee, Milwaukee, Wisconsin
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Let V ≠ 0 be a vector space of dimension n over a finite field of order qm for a prime q. Of course, GL(n, qm) denotes the group of -linear transformations of V. With few exceptions, GL(n, qm) is non-solvable. How large can a solvable subgroup of GL(n, qm) be? The order of a Sylow-q-subgroup Q of GL(n, qm) is easily computed. But Q cannot act irreducibly nor completely reducibly on V.

Suppose that G is a solvable, completely reducible subgroup of GL(n, qm). Huppert ([9], Satz 13, Satz 14) bounds the order of a Sylow-q-subgroup of G, and Dixon ([5], Corollary 1) improves Huppert's bound. Here, we show that |G| ≦ q3nm = |V|3. In fact, we show that

where

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 5 , 01 October 1982 , pp. 1097 - 1111
Copyright
Copyright © Canadian Mathematical Society 1982

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