Subset-conjugacy-determined subgroup

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This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.
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Definition

Suppose $H\leq K\leq G$ are groups. We say that $H$ is subset-conjugacy-determined in $K$, or that fusion of subsets of $H$ in $G$ is contained in $K$, if whenever $A,B\subseteq H$ and $g\in G$ is such that $gAg^{-1}=B$, there exists $k\in K$ such that $kak^{-1}=gag^{-1}$ for all $a\in A$.

If $H$ is subset-conjugacy-determined in itself relative to $G$, we say that $H$ is a subset-conjugacy-closed subgroup.

Relation with other properties

Weaker properties

• Conjugacy-determined subgroup
• Normal subset-conjugacy-determined subgroup
• Weak normal subset-conjugacy-determined subgroup

Related subgroup properties

• Subset-conjugacy-closed subgroup: A subgroup that is subset-conjugacy-determined in itself relative to the whole group.
• SCDIN-subgroup: A subgroup that is subset-conjugacy-determined in its normalizer, relative to the whole group.

Facts

• Center of pronormal subgroup is subset-conjugacy-determined in normalizer