# Why does $xH=H$ imply that $x$ is element of $H$

It's not much, but I'm still a little bit confused about cosets, and I noticed that

$$xH = H$$ implies $$x$$ to be an element of $$H$$.

Why does this make sense? Thank you!

• Try thinking of $H$ as a set, such as $\{ e,h_1,h_2,...\}$. – Mohammad Zuhair Khan May 14 '20 at 17:49
• $x\cdot e\in xH=H$ – A. Goodier May 14 '20 at 17:49
• – user750041 May 15 '20 at 5:31

If $$H$$ is a subgroup, then the identity element $$e\in H$$, so $$x=xe\in xH=H$$.
\begin{alignat}{1} &H=xH \Longrightarrow \\ &H\subseteq xH \Longrightarrow \\ &\forall h \in H, \exists h'\in H \mid h=xh' \Longrightarrow \\ &\forall h \in H, \exists h'\in H \mid x=hh'^{-1}\Longrightarrow \\ &x\in H \end{alignat}