Paper «On products of groups and indices not divisible by a given prime» published in Monatsh. Math.

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María José Felipe, Lev S. Kazarin, Ana Martínez-Pastor, and Víctor Sotomayor.
On products of groups and indices not divisible by a given prime.
Comm. Algebra, 193(4):811–827, 2020.

doi:10.1007/s00605-020-01446-z

Abstract

Let the group G = AB be the product of subgroups A and B, and let p be a prime. We prove that p does not divide the conjugacy class size (index) of each p-regular element of prime power order xAB if and only if G is p-decomposable, i.e. G=Op(G) × Op’(G).

2020 Mathematics Subject Classification: 20D40, 20E45, 20D20, 20D60

Keywords: Finite groups; products of groups; conjugacy classes; p-structure; prime graph; almost simple groups