Strongly closed 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

Definition with symbols

Let H \le K \le G be groups. Then, H is said to be strongly closed in K with respect to G if any G-conjugate of an element of H, which lies inside K, in fact lies inside H. In other words, for any g \in G:

gHg^{-1} \cap K \le H.

The term is typically used for the situation where K is a Sylow subgroup of G. The particular case can also be generalized to the notion of a strongly closed subgroup for a fusion system.

Relation with other properties

Weaker properties