A lower bound on the size of maximal abelian subgroups

2024/02/19 by Lewis, Mark L. · 2 citations
#20D25 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2402.12221

Abstract

Let G be a p-group for some prime p. Let n be the positive integer so that |G:Z(G)| = pⁿ. Suppose A is a maximal abelian subgroup of G. Let p^l = \rm max \|Z(C_G (g)):Z(G)| : g ∈ G ∖ Z(G)\, p^b = \rm max \|cl(g)| : g ∈ G ∖ Z(G) \, and p^a = |A:Z(G)|. Then we show that a ≥ n/(b+l).

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