Permutable subgroup

In mathematics, in the field of group theory, a quasinormal subgroup, or permutable subgroup, is a subgroup of a group that commutes (permutes) with every other subgroup with respect to the product of subgroups. The term quasinormal subgroup was introduced by Øystein Ore in 1937.

Two subgroups are said to permute (or commute) if any element from the first subgroup, times an element of the second subgroup, can be written as an element of the second subgroup, times an element of the first subgroup. That is, ${\displaystyle H}$ and ${\displaystyle K}$ as subgroups of ${\displaystyle G}$ are said to commute if HK = KH, that is, any element of the form ${\displaystyle hk}$ with ${\displaystyle h\in H}$ and ${\displaystyle k\in K}$ can be written in the form ${\displaystyle k'h'}$ where ${\displaystyle k'\in K}$ and ${\displaystyle h'\in H}$.

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