What is the significance of permutable subgroups? (and $X$-permutable subgroups?)

Question

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 ?

Answer 1

Short version: Permutability is a weakening of normality. Permutability is a common occurrence in solvable groups, but when specific groups are assumed to permute, one often gets very rich structural results on the entire group.

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Two subgroups permute if and only if their product is a subgroup: $HK=KH$ iff $HK$ is a subgroup.

Two subgroups are $X$-permutable if and only if some $X$-conjugates of them permute. This is particularly important if $H$ and $K$ were only chosen up to $X$-conjugacy in the first place (for instance if they are Sylow subgroups, Hall subgroups, system normalizers, Carter subgroups, Fischer subgroups, Sylow normalizers, or any other projector or injector).

A single subgroup $H$ is said to be $X$-permutable (with plain “permutable” meaning $X=\{1\}$) if for every subgroup $K$, $H$ and $K$ are $X$-permutable. Such subgroups are also called quasi-normal and share many properties of normality.

The study of normality includes normal, subnormal, pronormal, malnormal, permutable subgroups, and CAP subgroups, as well as local versions of each. Generally speaking, some of these properties are “subnormal-ish” and some are “pronormal-ish” with the only kind of subgroup that is both kinds are “normal-ish”. For instance, being a Sylow subgroup is kind a pronormal property, and being permutable is kind of a subnormal property, so it is not a surprise that a quasinormal Sylow subgroup is normal.

This example (to my mind) motivates a lot of the more arcane types of permutability studied. The proof is cheating: If $P$ is a quasinormal Sylow $p$-subgroup and $Q$ is a Sylow $p$-subgroup, then $PQ=QP$ is a subgroup, obviously a $p$-subgroup, and so $|PQ|=|P|$ and $Q=P$ is normal. To avoid such silly proofs, one has also sorts of Sylow permutability and semipermutability (with coprime subgroups) etc.

Answer 2

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.