The Collection Of Nonempty Subsets Of A Group Is A Monoid

Please Login to access more options.

Problem (The Collection Of Nonempty Subsets Of A Group Is A Monoid)

Let $G$ be a group, and consider the binary operation $HK$ on the collection $\mathscr{C}$ of nonempty subsets of $G$. Recall that $$HK=\{hk\mid h\in H, k\in K\}.$$ Prove the following:

• The set product $HK$ is indeed a binary operation on the collection $\mathscr{C}$ of nonempty subsets of $G$.
• The set product is associative, namely $(HK)L = H(KL)$. This proves that we have a semigroup.
• (Extra - We won't present this in class) The set $E=\{e\}$ is an identity element of $\mathscr{C}$, namely $EH=HE=H$ for any $H\in \mathscr{C}$. This proves that we have a monoid.

The following pages link to this page.