# Focal subgroup of a subgroup

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## Definition

Let $H$ be a subgroup of a group $G$. We define the focal subgroup of $H$ in the following equivalent ways:

• The subgroup generated by the left quotients of pairs of elements of $H$ which are conjugate in $G$.
• The subgroup generated by the right quotients of pairs of elements of $H$ which are conjugate in $G$.

We use the notation $\operatorname{Foc}_G(H)$ or $H^*_G$ for the focal subgroup of $H$ in $G$.

Note that the focal subgroup of $H$ in $G$ is contained within the commutator $[H,G]$, and contains the commutator $[H,H]$. In fact, we have the following string of inequalities:

$[H,H] \le \operatorname{Foc}_G(H) \le H \cap [H,G] \le H \cap [G,G]$.