permutable subgroup

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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 $K\leq G$ . We sometimes write $H\operatorname{per}G$ 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