Conjugate-permuting subgroups

This article defines a symmetric relation on the collection of subgroups inside the same group.

This is a variation of permuting subgroups|Find other variations of permuting subgroups |

Definition

Symbol-free definition

Two subgroups of a group are said to be conjugate-permuting if the following equivalent conditions are satisfied:

• Every conjugate of either permutes with every conjugate of the other.
• Every conjugate of one permutes with the other subgroup
• The first subgroup permutes with every conjugate of the other subgroup

Definition with symbols

Two subgroups $H$ and $K$ of a group $G$ are said to be conjugate-permuting if the following equivalent conditions are satisfied:

• $H^{g}$ permutes with $K^{h}$ for every $g,h\in G$
• $H^{g}$ permutes with $K$ for every $g\in G$
• $H$ permutes with $K^{g}$ for every $g\in G$

Relation with other relations

Stronger relations

• One is a seminormal subgroup and the other is a S-supplement of it

Weaker relations

• Permuting subgroups