Let $G$ be a cyclic group with prime power order and $H$ be a proper subgroup of $G$.
Problem: Any element in $G$ whose image in the cyclic quotient $G/H$ is a generator, is already a generator of $G$.
Approach: Let $G = \langle \sigma \rangle$ and $|G| = p^m$ be a prime power.
Because $H$ is a proper subgroup of G, there exists an $n$ which is not coprime to $p$ such that $H = \langle \sigma^n \rangle$. Then $G/H = \{ H, \sigma H, \dots, \sigma^{n-1}H \}$. The generators of $G/H$ are of the form $\sigma^k H$ where $1 \leq k \leq n-1$ and $\gcd(k,p)=1$.
How can I show that any lift $\sigma^{k+ln}$ of $\sigma^k H$ is a generator, i.e. $\gcd(k+ln,m)=1$?
Could you please help me with this problem? Any help is really appreciated.