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 \le K \le 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