Normal core

Definition

Let $G$ be a group and let $H$ be a subgroup of $G$. The normal core of $H$ in $G$ is the subgroup $\operatorname{Core}_G(H) = \bigcap_{g \in G} gHg^{-1}$.

Remarks

The core is a normal subgroup of $G$, and it contains each normal subgroup of $G$ that lies in $H$. The subgroup $H$ is normal exactly when $\operatorname{Core}_G(H) = H$. The core is the kernel of the action of $G$ on the left cosets of $H$ by left multiplication.

Explanations

L1 human

Some subgroups are not normal, but they still contain normal subgroups. The normal core is the biggest of these. You find it by keeping only the elements that lie in every conjugate of the subgroup.