On lattice properties of S-permutably embedded subgroups of finite soluble groups

Ezquerro, L. M. ; Gómez-Fernández, M. ; Soler-Escrivà, X.

Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 2, pp. 505-517

Résumé

In this paper we prove the following results. Let π be a set of prime numbers and G a finite π-soluble group. Consider U, V ≤ G and $H\in \mathrm{Hall}_{\pi}(G)$ such that $H\cap V \in \mathrm{Hall}_{\pi}(V)$ and $1\neq H\cap U\in \mathrm{Hall}_{\pi}(U)$. Suppose also $H \cap U$ is a Hall π-sub-group of some S-permutable subgroup of G. Then $H\cap U \cap V\in \mathrm{Hall}_{\pi}(U\cap V)$ and $\langle H\cap U, H\cap V \rangle\in \mathrm{Hall}_{\pi}(\langle U\cap V\rangle)$. Therefore,the set of all S-permutably embedded subgroups of a soluble group G into which a given Hall system Σ reduces is a sublattice of the lattice of all Σ-permutable subgroups of G. Moreover any two subgroups of this sublattice of coprimeorders permute.