Focal subgroup of a subgroup

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:

$[H,H]\leq \operatorname {Foc} _{G}(H)\leq H\cap [H,G]\leq H\cap [G,G]$.

Facts

• A subgroup $H\leq G$ for which $[H,H]=\operatorname {Foc} _{G}(H)$ is termed a subgroup whose focal subgroup equals its commutator subgroup. Any conjugacy-closed subgroup has this property. Further information: Conjugacy-closed implies focal subgroup equals commutator subgroup
• A subgroup $H\leq G$ for which $\operatorname {Foc} _{G}(H)=H\cap [G,G]$ is termed a subgroup whose focal subgroup equals its intersection with the commutator subgroup. Any Sylow subgroup has this property. Further information: Focal subgroup theorem