Please Login to access more options.
Problem 80 (When Does $H=Ha$)
Let $H$ be a nonempty subset of a group $G$. Prove that $H$ is a subgroup of $G$ if and only if $Ha=H$ for every $a\in H$.
Click if you want a hint.
If you assume $H$ is a subgroup, pick $a\in H$ and then prove that $H\subseteq Ha$ and $Ha\subseteq H$.
If you assume that $Ha=H$ for every $a\in H$, then you must prove $H$ is a subgroup. You'll need to rely on the fact that if $b,c\in H$, then we know $H=Hb$ and $H=Hc$ as sets. One of these should get you closure pretty quickly. If you let $b\in H$, then why does $H=Hb$ force the existence of $h\in H$ such that $b=hb$, and as such why does this mean $e\in H$. A similar argument should get you the fact that $b^{-1}\in H$ (as $e\in H$ forces $e=bh$ for some $h\in H$).
The following pages link to this page.