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George Glauberman

Abelian subgroups of small index in finite p-groups

Suppose p is a prime, S is a finite p -group, and A is an abelian subgroup of S. Does S possess a normal abelian subgroup of the same order as A? J. Alperin showed that the answer is negative in general. However, by using methods of algebraic group theory, he and the author showed that the answer is affirmative if |A| = p ⁿ, where n is ‘small’ relative to p.

In this paper, we prove the dual result that the answer is affirmative if the index |S : A| is ‘small’ relative to p. We also prove a correspondence between the abelian subgroups of a p -group of nilpotence class at most p and the abelian subgroups of a related p -group of nilpotence class at most two.

Journal of Group Theory, Walter de Gruyter

Print ISSN: 1433-5883
Volume: 8, 09/2005
Seiten: 539 - 560

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