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On products of groups and indices not divisible by a given prime

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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 \(x\in A\cup B\) if and only if G is p-decomposable, i.e. \(G=O_p(G) \times O_{p'}(G)\).

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Correspondence to Ana Martínez-Pastor.

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Communicated by John S. Wilson.

Dedicated to the memory of Carlo Casolo

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Research supported by Proyecto PGC2018-096872-B-I00 from the Ministerio de Ciencia, Innovación y Universidades, Spain, and FEDER. The second author is also supported by Project VIP-008 of Yaroslavl P. Demidov State University and the third author by Proyecto PROMETEO/2017/057 from the Generalitat Valenciana, Spain.

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Felipe, M.J., Kazarin, L.S., Martínez-Pastor, A. et al. On products of groups and indices not divisible by a given prime. Monatsh Math 193, 811–827 (2020).

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