Conjugate-permutability is conjugate-join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., conjugate-permutable subgroup) satisfying a subgroup metaproperty (i.e., conjugate-join-closed subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
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Statement

Statement with symbols

If $H$ is a conjugate-permutable subgroup of a group $G$, and $S$ is any subset of $G$, then the subgroup:

$H^{S}:=\langle H^{g}\mid g\in S\rangle$

is also a conjugate-permutable subgroup of $G$.

Related facts

Similar facts

• 2-subnormality is conjugate-join-closed
• Normality is strongly join-closed
• Permutability is strongly join-closed

Applications

• Maximal conjugate-permutable implies normal