For a group G, denote by ω(G) the number of conjugacy classes of normalizers of subgroups of G. Clearly, ω(G) = 1 if and only if G is a Dedekind group. Hence if G is a 2-group, then G is nilpotent of class ≤ 2 and if G is a p-group, p > 2, then G is abelian. We prove a generalization of this. Let G be a finite p-group with ω(G) ≤ p + 1. If p = 2, then G is of class ≤ 3; if p > 2, then G is of class ≤ 2.

p-Groups with few Conjugacy Classes of Normalizers

SICA, Carmela;TOTA, Maria
2013

Abstract

For a group G, denote by ω(G) the number of conjugacy classes of normalizers of subgroups of G. Clearly, ω(G) = 1 if and only if G is a Dedekind group. Hence if G is a 2-group, then G is nilpotent of class ≤ 2 and if G is a p-group, p > 2, then G is abelian. We prove a generalization of this. Let G be a finite p-group with ω(G) ≤ p + 1. If p = 2, then G is of class ≤ 3; if p > 2, then G is of class ≤ 2.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11386/3991454
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