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Generating Finite Groups with Conjugates of a Subgroup, II

✍ Scribed by Paul Flavell

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
298 KB
Volume
232
Category
Article
ISSN
0021-8693

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✦ Synopsis

Suppose that H is a subgroup of a finite group G and that G is generated by the conjugates of H. In this paper, we consider the following question: when can G be generated by two conjugates of H?

We began the study of this question in [2]. In order to discuss the results proved in [2] and in this paper, we use the following notation. The chain length of H in G is defined by

If cl G H = 1 then H is maximal in G and the answer to the question is yes. In [2] we considered the case cl G H = 2. We proved that the answer is yes, unless G has a very restricted structure. Note that there is no loss in factoring out by H G , the largest normal subgroup of G contained in H and hence assuming H G = 1. In [2] we proved: Theorem A. Let H be a subgroup of a finite group G such that G = H G and H G = 1. Suppose that cl G H = 2 and that G cannot be generated by two conjugates of H. Then there exists a prime p and a faithful irreducible

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## Abstract Let __X__ be a nonempty subset of a group __G__. We call a subgroup __A__ of __G__ an __X__~__m__~‐__semipermutable__ subgroup of __G__ if __A__ has a minimal supplement __T__ in __G__ such that for every maximal subgroup __M__ of any Hall subgroup __T__~1~ of __T__ there exists an elem