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 $\Gamma$?
On
For another proof (though it only answers the title as the $n$ we get is not $[G:H]$)
Let $H$ be a finite index subgroup. Then $G$ acts on $G/H$ by left translation, and so we have a morphism $\rho : G\to \mathfrak{S}G/H$.
Its kernel $K$ is contained in $H$ : indeed if $x\in K$, then $H= \rho(x)(H)= xH$, so $x\in H$. Moreover, $K$ is normal (it's a kernel !), and it has finite index in $G$ (because $G/K\simeq \mathrm{Im}\rho \subset \mathfrak{S}G/H\simeq \mathfrak{S}_{|G/H|}$).
Thus if $x\in G, x^n\in K$ for some $n$, thus $x^n\in H$ for some $n$. Note, however, that $n$ is not necessarily $[G:H]$; and the proof I gave only gives the bound $[G:H]!$ for $n=[G:K]$.
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 $g^{a-b} \in H$.