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

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

Bollettino dell'Unione Matematica Italiana (2005)

  • Volume: 8-B, Issue: 2, page 505-517
  • ISSN: 0392-4041

Abstract

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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 ∈ Hall π ( G ) such that H ∩ V ∈ Hall π ( V ) and 1 ≠ H ∩ U ∈ Hall π ( U ) . Suppose also H ∩ U is a Hall π-sub-group of some S-permutable subgroup of G. Then H ∩ U ∩ V ∈ Hall π ( U ∩ V ) and 〈 H ∩ U , H ∩ V 〉 ∈ Hall π ( 〈 U ∩ V 〉 ) . 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.

How to cite

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Ezquerro, L. M., Gómez-Fernández, M., and Soler-Escrivà, X.. "On lattice properties of S-permutably embedded subgroups of finite soluble groups." Bollettino dell'Unione Matematica Italiana 8-B.2 (2005): 505-517. <http://eudml.org/doc/194694>.

@article{Ezquerro2005,
abstract = {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 $1\ne H\cap U\in \mathrm \{Hall\}_\{\pi \}(U)$. 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.},
author = {Ezquerro, L. M., Gómez-Fernández, M., Soler-Escrivà, X.},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {finite -soluble groups; Hall subgroups; -permutable subgroups; permutably embedded subgroups; Hall systems; lattices of subgroups},
language = {eng},
month = {6},
number = {2},
pages = {505-517},
publisher = {Unione Matematica Italiana},
title = {On lattice properties of S-permutably embedded subgroups of finite soluble groups},
url = {http://eudml.org/doc/194694},
volume = {8-B},
year = {2005},
}

TY - JOUR
AU - Ezquerro, L. M.
AU - Gómez-Fernández, M.
AU - Soler-Escrivà, X.
TI - On lattice properties of S-permutably embedded subgroups of finite soluble groups
JO - Bollettino dell'Unione Matematica Italiana
DA - 2005/6//
PB - Unione Matematica Italiana
VL - 8-B
IS - 2
SP - 505
EP - 517
AB - 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 $1\ne H\cap U\in \mathrm {Hall}_{\pi }(U)$. 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.
LA - eng
KW - finite -soluble groups; Hall subgroups; -permutable subgroups; permutably embedded subgroups; Hall systems; lattices of subgroups
UR - http://eudml.org/doc/194694
ER -

References

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  10. EZQUERRO, L. M. - SOLER-ESCRIV, X., Some permutability properties of F -hypercentrally embedded subgroups of finite groups. Preprint (to appear in J. Algebra), 2002. Zbl1107.20018MR1980698
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