Let $G$ be a group, not necessarily finite. If $H$ is a normal subgroup of $G$ of a finite index, say $(G:H)=n$, then for every $g\in G$ we have $g^n\in H$. Does this statement remain valid if do not assume $H$ to be normal?
In particular let $SL_2(\mathbb Z)$ be the modular group, and let $\Gamma\subset SL_2(\mathbb Z)$ be a subgroup of a finite index. Does there exists a positive integer $\ell$ such that $\begin{pmatrix}1&1\\0 & 1\end{pmatrix}^\ell$ lies in $\Gamma$?
Best Answer
Yes.
The set $\{ H, gH, g^2H, \dots , g^nH \}$ has $n+1$ elements, so that two of them are equal ( $H$ has only $n$ right cosets).
From $$g^aH=g^bH$$ it follows that $g^{a-b} \in H$.