# permutable subgroup

Let $G$ be a group.
A subgroup^{} $H$ of $G$ is said to be *permutable*
if it permutes with all subgroups of $G$,
that is, $KH=HK$ for all $K\le G$.
We sometimes write $H\mathrm{per}G$
to indicate that $H$ is a permutable subgroup of $G$.

Permutable subgroups were introduced by Øystein Ore (http://planetmath.org/OysteinOre),
who called them *quasinormal* subgroups.

Normal subgroups^{} are clearly permutable.

Permutable subgroups are ascendant. This is a result of Stonehewer[1], who also showed that in a finitely generated group, all permutable subgroups are subnormal.

## References

- 1 Stewart E. Stonehewer, Permutable subgroups of infinite groups, Math. Z. 125 (1972), 1–16. (This paper is http://gdz.sub.uni-goettingen.de/dms/resolveppn/?GDZPPN002410435available from GDZ.)

Title | permutable subgroup |
---|---|

Canonical name | PermutableSubgroup |

Date of creation | 2013-03-22 16:15:47 |

Last modified on | 2013-03-22 16:15:47 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20E07 |

Synonym | quasinormal subgroup |

Synonym | quasi-normal subgroup |

Defines | permutable |

Defines | quasinormal |

Defines | quasi-normal |