# Strongly closed subgroup

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

### 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.