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 \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.

Relation with other properties

Weaker properties

Related subgroup properties

Facts