Group theory MOC

Core of a subgroup

Let 𝐺 be a group, 𝐻 ≤𝐺 be a subgroup, and 𝑆 ⊆𝐺 be a subset. The core of 𝐻 under 𝑆 is the intersection of the conjugates of 𝐻 under 𝑆, group i.e.

Core𝑆⁡𝐻=⋂𝑠∈𝑆𝑠𝐻𝑠−1

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

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