[[Group theory MOC]]
# Core of a subgroup

Let $G$ be a group, $H \leq G$ be a subgroup, and $S \sube G$ be a subset.
The **core** of $H$ under $S$ is the intersection of the conjugates of $H$ under $S$, #m/def/group  i.e.
$$
\begin{align*}
\opn{Core}_{S}H = \bigcap_{s \in S} sHs^{-1}
\end{align*}
$$
In particular, if $S = G$ one gets the **normal interior** $H^\circ$ of $H$, the maximal normal subgroup $H^\circ \trianglelefteq G$ contained within $H$.


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