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 conjugacy-determined in $K$ relative to $G$, or that $K$ contains fusion of elements of $H$ in $G$, if two elements of $H$ are conjugate in $K$ if and only if they are conjugate in $G$.

If $H$ is conjugacy-determined in itself relative to $G$, $H$ is termed a conjugacy-closed subgroup of $G$.

Relation with other properties

Stronger properties

• Subset-conjugacy-determined subgroup