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Groups with a Bounded Number of Conjugacy Classes of Non-normal Subgroups

✍ Scribed by Roberta La Haye; Akbar Rhemtulla

Publisher
Elsevier Science
Year
1999
Tongue
English
Weight
169 KB
Volume
214
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let Ξ½ G denote the number of conjugacy classes of non-normal subgroups of a group G We prove that if G is a finite group and Ξ½ G = 0 then there is a cyclic subgroup C of prime power order contained in the centre of G such that the order of G/C is a product of at most Ξ½ G + 1 primes. We also obtain a bound in the opposite direction, thus obtaining a criterion for a group to have a bounded number of conjugacy classes of non-normal subgroups. These results extend to infinite groups, with the subgroup C being the infinite PrΓΌfer p-group, but only when G has finitely many non-normal subgroups. This is to be expected because of the existence of monsters of the type constructed by S. V. Ivanov and A. Yu. Ol'shanskii. The structure of groups with infinitely many non-normal subgroups falling into finitely many conjugacy classes is also studied.

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