The Correspondence Theorem

Question

Let f be a homomorphism from G onto H with kernel K: f:Gk--->H. If S is any subgroup of H, let S*={x∈G: f(x) ∈ S}. Prove: S ≅S* / K

Answer 1

Hint: For a group $G$ and its normal subgroup $N$, the elements of the quotient group $G/N$ are basically those of $G$, but all $n\in N$ is identified with $1$, and as all the consequences of these we have that $g$ and $g'$ are equal as elements of $G/N$ iff $g'=gn$ for some $n\in N$.

Then apply the same to the subgroup $S$, and the required isomorphism is basically given by $f$.