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On theta pairs for a maximal subgroup

Authors: N. P. Mukherjee and Prabir Bhattacharya
Journal: Proc. Amer. Math. Soc. 109 (1990), 589-596
MSC: Primary 20D10; Secondary 20D25
MathSciNet review: 1015683
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Abstract: For a maximal subgroup $ M$ of a finite group $ G$, a $ \Theta $-pair is any pair of subgroups $ (C,D)$ of $ G$ such that (i) $ D \triangleleft G,D \subset C$, (ii) $ \left\langle {M,C} \right\rangle = G,\left\langle {M,D} \right\rangle = M$ and (iii) $ C/D$ has no proper normal subgroup of $ G/D$. A natural partial ordering is defined on the family of $ \Theta $-pairs. We obtain several results on the maximal $ \Theta $-pairs which imply $ G$ to be solvable, supersolvable, and nilpotent.

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Keywords: Solvable, supersolvable, nilpotent
Article copyright: © Copyright 1990 American Mathematical Society

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