Tour:Permuting subgroups

This article adapts material from the main article: permuting subgroups

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
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WHAT YOU NEED TO DO:

• Read, and understand, the many equivalent definitions of permuting subgroups. (You can omit definition (4) for now if you are not familiar with the notation).
• Try to prove the equivalence of definitions (this is the next tour page). You can omit proving the equivalence of definition (4).

Definition

Definition with symbols

Two subgroups $H$ and $K$ of a group $G$ are termed permuting subgroups if the following equivalent conditions hold:

1. $HK=KH$
2. $HK$ (the product of subgroups) is a subgroup
3. Given elements $h$ in $H$ and $k$ in $K$, there exist elements $k'$ in $K$ and $h'$ in $H$ such that $hk=k'h'$. In other words, $HK\subseteq KH$.
4. $[H,K]\subseteq HK$. In other words, the commutator of $H$ and $K$ is contained in their product.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of permuting subgroups

This page is part of the Groupprops guided tour for beginners (Jump to beginning of tour)
PREVIOUS: Product of subgroups| UP: Introduction five (beginners)| NEXT: Equivalence of definitions of permuting subgroups
General instructions for the tour | Pedagogical notes for the tour | Pedagogical notes for this part