$(\subset)$ Take an arbitrary element $g\in f^{-1}(f(H))$. Then we have $f(g)\in f(H)$.
It follows that there exists $h\in H$ such that $f(g)=f(h)$.
Since $f$ is a group homomorphism, we obtain
\[f(h^{-1}g)=e’,\]
where $e’$ is the identity element of the group $G’$.

This implies that $h^{-1}g\in \ker(f)=N$, hence we have
\begin{align*}
g\in hN\subset HN.
\end{align*}
Therefore we have $f^{-1}(f(H)) \subset HN$.

$(\supset)$ On the other hand, let $g\in HN$ be an arbitrary element.
Then we can write $g=hn$ with $h \in H$ and $n\in N$.
We have
\begin{align*}
f(g)&=f(hn)=f(h)f(n)\\
&=f(h)e’=f(h)\in f(H)
\end{align*}
since $f$ is a group homomorphism and $f(n)=e’$.
Thus we have
\[g\in f^{-1}(f(H))\]
and $f^{-1}(f(H)) \supset HN$.
Therefore, putting the two continents together gives
\[f^{-1}(f(H))=HN\]
as required.

Surjective Group Homomorphism to $\Z$ and Direct Product of Abelian Groups
Let $G$ be an abelian group and let $f: G\to \Z$ be a surjective group homomorphism.
Prove that we have an isomorphism of groups:
\[G \cong \ker(f)\times \Z.\]
Proof.
Since $f:G\to \Z$ is surjective, there exists an element $a\in G$ such […]

Group Homomorphisms From Group of Order 21 to Group of Order 49
Let $G$ be a finite group of order $21$ and let $K$ be a finite group of order $49$.
Suppose that $G$ does not have a normal subgroup of order $3$.
Then determine all group homomorphisms from $G$ to $K$.
Proof.
Let $e$ be the identity element of the group […]

Subgroup of Finite Index Contains a Normal Subgroup of Finite Index
Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.
Proof.
The group $G$ acts on the set of left cosets $G/H$ by left multiplication.
Hence […]

A Group Homomorphism is Injective if and only if Monic
Let $f:G\to G'$ be a group homomorphism. We say that $f$ is monic whenever we have $fg_1=fg_2$, where $g_1:K\to G$ and $g_2:K \to G$ are group homomorphisms for some group $K$, we have $g_1=g_2$.
Then prove that a group homomorphism $f: G \to G'$ is injective if and only if it is […]

The Preimage of a Normal Subgroup Under a Group Homomorphism is Normal
Let $G$ and $G'$ be groups and let $f:G \to G'$ be a group homomorphism.
If $H'$ is a normal subgroup of the group $G'$, then show that $H=f^{-1}(H')$ is a normal subgroup of the group $G$.
Proof.
We prove that $H$ is normal in $G$. (The fact that $H$ is a subgroup […]

Abelian Normal subgroup, Quotient Group, and Automorphism Group
Let $G$ be a finite group and let $N$ be a normal abelian subgroup of $G$.
Let $\Aut(N)$ be the group of automorphisms of $G$.
Suppose that the orders of groups $G/N$ and $\Aut(N)$ are relatively prime.
Then prove that $N$ is contained in the center of […]

Group of $p$-Power Roots of 1 is Isomorphic to a Proper Quotient of Itself
Let $p$ be a prime number. Let
\[G=\{z\in \C \mid z^{p^n}=1\} \]
be the group of $p$-power roots of $1$ in $\C$.
Show that the map $\Psi:G\to G$ mapping $z$ to $z^p$ is a surjective homomorphism.
Also deduce from this that $G$ is isomorphic to a proper quotient of $G$ […]

A Group Homomorphism that Factors though Another Group
Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$.
Define a map $\bar{f}:H\to K$ as follows.
For each […]