What is the significance of permutable subgroups? (and $X$-permutable subgroups?) Let $G$ be a group and $H$, $K$, $X$ be subgroups of $G$. We say $H$, $K$ are permutable if $HK=KH$.
or we say $H, K$ are X-permutable if $‎\exists x, x\in X$ such that $H^{x}K=KH^{x}.$
Why are permutable subgroups important to us ?

Permutable subgroups are essential in the study of solvable groups.

Definition. A Sylow system of a group $G$ is a collection of pairwise permutable Sylow subgroups of $G$, one for each prime divisor of $|G|$.
Theorem. (Hall) A group is solvable if and only if it has a Sylow system.

For example, if $S_pS_q=S_qS_p$ for distinct primes $p$ and $q$, we say that $S_pS_q$ is a Hall $\{p,q\}$-subgroup.  Hall subgroups are generalizations of Sylow subgroups for multiple primes, and since they always exist in solvable groups (by Hall), it is to this class of groups in which Hall subgroups are most crucially studied.
