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On a class of finite soluble groups
Published online by Cambridge University Press: 17 April 2009
Abstract
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Let the group T be the direct product of groups Si (i = 1, … r) where for a given group Ai, Si is the direct product of ni factors Ai × Ai × … × Ai. Let B be a group that has a faithful permutation representation Γi of degree ni. (i = 1, …, r). Consider G, the split extension of T by B defined by letting B act on T as follows.
Each Si is normal in G. If 
and b ∈ B then 
where 
. It is proved that if T is an M-group and all subgroups of B are M-groups, then G is an M-group. This is a generalisation of a result of Gary M. Seitz, Math. Z. 110 (1969), 101–122, who proved the particular case where r = 1 and Γ1 is the regular representation of B.
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- Research Article
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- Copyright © Australian Mathematical Society 1975
References
[1]Curtis, Charles W., Reiner, Irving, Representation theory of finite groups and associative algebras (Pure and Applied Mathematics, 11. Interscience [John Wiley & Sons], New York, London, 1962).Google Scholar
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[5]Winter, David L. and Murphy, Paul F., “Groups all of whose subgroups are M-groups”, Math. Z. 124 (1972), 73–78.Google Scholar
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