Inverse image of homomorphism

Question

If $\phi : G \rightarrow G'$ is a homomorphism, and $K'$ is a subgroup of $G'\cap\phi[G]$, $\phi^{-1}[K']$ is a subgroup of $G$. Why must $K'$ be a subgroup of $G'\cap\phi[G]$, and not only $G'$ or only $\phi[G]$? I can't find an example where it wouldn't work with only either of the two subgroups. What is a thought process to come up with an example?

Answer 1

First of all, since $\phi[G]\subset G',$ $G'\cap\phi[G]=\phi[G].$

So, your statement boils down to: if $K'$ is a subgroup of $\phi[G]$ then $\phi^{-1}[K']$ is a subgroup of $G$.

But we also have: if $H$ is a subgroup of $G'$ then $\phi^{-1}[H]$ is a subgroup of $G$.

• This is more general, because every subgroup of $\phi[G]$ is a subgroup of $G'$.
• This results from the previous particular case applied to $K':=H\cap\phi(G),$ because then, $$\phi^{-1}[K']=\phi^{-1}[H\cap\phi(G)]=\phi^{-1}[H]\cap\phi^{-1}[\phi(G)]=\phi^{-1}[H]\cap G=\phi^{-1}[H].$$