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Finite Groups with Given Weakly σ-Permutable Subgroups

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Abstract

Let G be a finite group and let σ = {σ i | i ∈ I} be a partition of the set of all primes P. A set ℋ of subgroups of G is said to be a complete Hall σ-set of G if each nonidentity member of ℋ is a Hall σ i -subgroup of G and ℋ has exactly one Hall σ i -subgroup of G for every σ i ∈ σ(G). A subgroup H of G is said to be σ-permutable in G if G possesses a complete Hall σ-set ℋ such that HAx = AxH for all A ∈ ℋ and all x ∈ G. A subgroup H of G is said to be weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H σG , where H σG is the subgroup of H generated by all those subgroups of H which are σ-permutable in G. We study the structure of G under the condition that some given subgroups of G are weakly σ-permutable in G. In particular, we give the conditions under which a normal subgroup of G is hypercyclically embedded. Some available results are generalized.

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Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, P. R. China

    C. Cao, Z. Wu & W. Guo

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  1. C. Cao
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  2. Z. Wu
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  3. W. Guo
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Correspondence to C. Cao.

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Original Russian Text Copyright © 2018 Cao C., Wu Z., and Guo W.

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 59, No. 1, pp. 197–209, January–February, 2018; DOI: 10.17377/smzh.2018.59.117

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Cao, C., Wu, Z. & Guo, W. Finite Groups with Given Weakly σ-Permutable Subgroups. Sib Math J 59, 157–165 (2018). https://doi.org/10.1134/S0037446618010172

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  • DOI: https://doi.org/10.1134/S0037446618010172

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