QPL-group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]

History

Origin

The term was introduced by William E. Deskins in his paper On Quasinormal subgroups of finite groups, in an attempt to try to characterize what additional information could be gathered about the permutable subgroups of a group under the further condition that the intersection of permutable subgroups is permutable.

The term quasinormal subgroup is an old term for permutable subgroup.

Definition

Symbol-free definition

A finite group is said to be a QPL-group if every subgroup of it of prime power order satisfies the condition that the intersection of any two permutable subgroups in it is permutable.

Facts

Any permutable subgroup of a QPL-group is an Abelian quotient-by-core subgroup, viz its quotient by its normal core is Abelian.

References

  • On quasinormal subgroups of finite groups by William E. Deskins, Math. Zeitschr. 82, 125-132 (1963)