# On the Lattice of all \({{\mathcal {H}}}\)-Permutable Subgroups of a Finite Group

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## Abstract

Let *G* be a finite group and \(\sigma =\{\sigma _{i} | i\in I\}\) is a partition of all primes \(\mathbb {P}\). A set \({{{\mathcal {H}}}}\) of subgroups of *G* is called a complete Hall \(\sigma \)-set of *G* if every member \(\ne 1\) of \({{{\mathcal {H}}}}\) is a Hall \(\sigma _{i}\)-subgroup of *G* for some \(\sigma _{i} \in \sigma \) and \({\mathcal {H}}\) contains exactly one Hall \(\sigma _{i}\)-subgroup of *G* for every *i* such that \(\sigma _{i}\in \sigma (G)\); a \(\sigma \)-basis of *G* if \({{{\mathcal {H}}}}\) is a complete Hall \(\sigma \)-set of *G* such that each of the subgroups \(A, B\in {{{\mathcal {H}}}}\) permutable. For any set \({{\mathcal {H}}}\) of subgroups of *G*, we write \({{{\mathcal {P}}}}({{\mathcal {H}}})\) to denote the set of all subgroups *A* of *G* such that *A* permutes with all subgroups \(B\in {{{\mathcal {H}}}}\), that is, \(AB=BA\); \({{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)\) to denote the set of all subgroups *A* of *G* with \(A\in {{{\mathcal {P}}}}({{\mathcal {H}}})\) and soluble \(A^{G}/A_{G}\). We prove that the set \({{{\mathcal {P}}}} ({{\mathcal {H}}})\) and the set \({{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)\) form sublattices of the lattice \({{{\mathcal {L}}}}(G)\) of all subgroups of *G* for each \(\sigma \)-basis \({{\mathcal {H}}}\) of *G*. We prove also that if all members of \({{\mathcal {H}}}\) are nilpotent, then the lattice \({{{\mathcal {L}}}}_{{{\mathcal {H}}}{\mathfrak {S}}}(G)\) is modular if and only if each of the two subgroups \(A, B \in {{{\mathcal {L}}}}_{{{{\mathcal {H}}}}{\mathfrak {S}}}(G)\) are permutable.

## Keywords

Finite group \(\sigma \)-Basis Permutable subgroups Subgroup lattice Modular lattice## Mathematics Subject Classification

20D10 20D30 20E15## Notes

### Acknowledgements

The authors are deeply grateful to the helpful suggestions of the referee.

### Compliance with Ethical Standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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