Here, I have the following homework:
Let $G$ is a finite $p-$group and let $H$ be a subgroup of it such that $HG'=G$. Prove that $H=G$ ($G'$ is the commutator subgroup).
I have tried to show that $G\subseteq H$ by taking an element in $G$ but this way seems to be weak here. Is it possible that this exercise is printed mistakenly? Thank you friends.
It suffices to prove that $G'$ is a subgroup of $\Phi(G)$, the Frattini subgroup of $G$. In the case of $p$-groups, you can do this by showing that $G/\Phi(G)$ is elementary abelian. (This is actually true for nilpotent groups in general, in which case $G/\Phi(G)$ just has to be abelian, though not necessarily elementary.)