permutable subgroup

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permutable subgroups

Let $G$ be a group.A subgroup $H$ of $G$ is said to be permutableif 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.)

单词	PermutableSubgroup
释义	permutable subgroup \\PMlinkescapephrase permutable subgroups Let $G$ be a group.A subgroup $H$ of $G$ is said to be permutableif 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.)
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