On the $\sigma_i$-length of a finite $\sigma$-soluble group

N. S. Kosenok ; V. M. Selkin

Problemy fiziki, matematiki i tehniki, no. 4 (2020), pp. 95-97.

Résumé

Let $\sigma=\{\sigma_i\mid i\in I\}$ be some partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. $G$ is said to be $\sigma$-soluble if every chief factor $H/K$ of $G$ is a $\sigma_i$-group for some $i=i(H/K)$. We prove the following Theorem. (i) If $G$ is $\pi$-separable, $H$ is a nilpotent Hall $\pi$-subgroup and $E$ a $\pi$-complement of $G$ such that $EX=XE$ for some subgroup $X$ of $H$ such that $H'\leqslant X\leqslant \Phi(H)$, then $l_\pi(G)\leqslant1$. (ii) If $G$ is $\sigma$-soluble and $\{H_1,\dots, H_t\}$ is a Wielandt $\sigma$-basis of $G$ such that $H_i$ permutes with $H_j$ for all $i$, $j$, then $l_{\sigma_i}(G)\leqslant 1$ for all $i$. (iii) If $G$ is $\sigma$-soluble and $\{H_1,\dots, H_t\}$ is a Wielandt $\sigma$-basis of $G$ such that $H_i$ permutes with $\Phi(H_j)$ for all $i$, $j$, then $l_{\sigma_i}(G)\leqslant 1$ for all $i$. (iv) If $l_\pi(G)\leqslant 1$, then $QX=XQ$ each characteristic subgroup $X$ of $H$ and any Sylow subgroup $Q$ of $G$ such that $HQ=QH$. (v) If $G$ is $\sigma$-soluble with $l_{\sigma_i}\leqslant 1$ for all $i$ and $\{H_1,\dots, H_t\}$ is a $\sigma$-basis of $G$, then each characteristic subgroup of $H_i$ permutes with each characteristic subgroup of $H_j$.

Détail

Keywords: finite group, $\pi$-separable group, $\pi$-length, Hall subgroup.
Mots-clés : $\sigma$-soluble group

Bibliographie

[1] A.N. Skiba, “A generalization of a Hall theorem”, J. Algebra and its Application, 15:5 (2016), 1650085 | DOI | MR | Zbl

[2] A.N. Skiba, “Some characterizations of finite $\sigma$-soluble $P\sigma T$-groups”, J. Algebra, 495 (2018), 114–129 | DOI | MR | Zbl

[3] A.N. Skiba, “On sublattices of the subgroup lattice defined by formation Fitting sets”, J. Algebra, 550 (2020), 69–85 | DOI | MR | Zbl

[4] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York–Heidelberg–Berlin, 1982 | MR | Zbl

[5] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin–Heidelberg–New York, 1967 | MR | Zbl

[6] L.A. Shemetkov, A.N. Skiba, Formations of Algebraic Systems, Nauka, M., 1989 | MR | Zbl

[7] K. Doerk, T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin–New York, 1992 | MR