Core of a subgroup

Let $𝐺$ be a group, $𝐻 \leq 𝐺$ be a subgroup, and $𝑆 \subseteq 𝐺$ be a subset. The core of $𝐻$ under $𝑆$ is the intersection of the conjugates of $𝐻$ under $𝑆$ , #m/def/group i.e.

C o r e 𝑆 𝐻 = ⋂ 𝑠 \in 𝑆 𝑠 𝐻 𝑠 - 1

In particular, if $𝑆 = 𝐺$ one gets the normal interior $𝐻 \circ$ of $𝐻$ , the maximal normal subgroup $𝐻 \circ ⊴ 𝐺$ contained within $𝐻$ .

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