# Subset-conjugacy-determined subgroup

## Contents

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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.
View other such properties

## Definition

Suppose $H \le K \le G$ 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.