I want to show for any group $G$ that $[G,G]\cap Z(G)\subseteq \Phi(G)$.
But I don't really know why that works. I looked at the definition of the different groups: $[G,G]=\langle[a,b] | a,b\in G\rangle$, $[a,b]=aba^{-1}b^{-1}$. So when the elements in the intersection are the $a,b\in G$ s.t $[a,b]=e$.
The thing is that all the usefull Lemma's & co. only for finite $G$ are, and I don't know how to show that the intersection must lay in $\Phi(G)$.
I hope someone is willing to give some hints :)