I am working on some elementary group theory problems and one of them states:
"Show that an infinite group $G$ has to contain a non-trivial subgroup, i.e. a subgroup $\neq G$, {$e$}."
My process:$$G:\{a,b,...,e,a^{-1},b^{-1}...\}$$
So a possible subgroup could be $$G_{s}: \{a,a^{-1},e\}$$ Since $$a*a^{-1}=e$$ $$a*e=a$$ $$a^{-1}*e=a^{-1}$$ are all contained within the subgroup.
Is this approach valid? I feel like I'm missing something...
EDIT: Totally forgot about $a*a$, could anyone give me any hints on how to approach this problem?