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On the Intersection of a Family of Maximal Subgroups Containing the Sylow Subgroups of a Finite Group

Published online by Cambridge University Press:  20 November 2018

N. P. Mukherjee  and
Prabir Bhattacharya
N. P. Mukherjee
Affiliation:
Jawaharlal Nehru University, New Delhi, India
Prabir Bhattacharya
Affiliation:
University of Nebraska – Lincoln, Lincoln, Nebraska
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Given a finite group G, the Frattini subgroup of G, Φ(G) is defined to be the intersection of all the maximal subgroups of G. Of late there have been several attempts to consider generalizations of Φ(G). For example, Gaschutz [7] and Rose [13] have investigated the intersection of all non-normal, maximal subgroups of a finite group. Deskins [6] has discussed the intersection of the family of maximal subgroups of a finite group whose indices are co-prime to a given prime. In [4-5, 12] we have considered the investigation of the family of all maximal subgroups of a finite group whose indices are composite and co-prime to a given prime. We have obtained several results about the family . In this paper which is a sequel to [4] we prove some further results about this family indicating the interesting role it plays especially when G is solvable or p-solvable. First we recall the main definition from [4].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 2 , 01 April 1988 , pp. 352 - 359
Copyright
Copyright © Canadian Mathematical Society 1988

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