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A class of modules over a locally finite group III

Published online by Cambridge University Press:  17 April 2009

B. Hartley
B. Hartley
Affiliation:
Mathematics Institute, University of Warwick, Coventry, England; Department of Mathematics, University of Wisconsin, Madison, Wisconsin, USA.
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Let G be a locally finite group, k a field of characteristic p ≥ 0 and V a right kG-module. We say that V is an -module over kG, if each p′-subgroup H of G contains a finite subgroup F with the same fixed points as H in V. (By convention, 0′ is taken as the set of all primes.) Such modules arise as elementary abelian section of -groups, a class of locally finite groups similar in many ways to the class of finite soluble groups.

The main theorem is that if V is an -module over kG with trivial Frattini submodule, and G is almost abelian, then every composition factor of V is complemented. This is a crucial ingredient in Tomkinson's theory of prefrattini subgroups in a certain subclass of . An example is given to show that the theorem breaks down for metabelian G. This leads to an example of a -group in which there are no analogues of prefrattini subgroups - the first situation where one of the standard conjugacy classes of subgroups of finite soluble groups has no decent analogue in the whole class

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 14 , Issue 1 , February 1976 , pp. 95 - 110
Copyright
Copyright © Australian Mathematical Society 1976

References

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