First of all, since $A \triangleleft G$, the product $AB$ is a subgroup of $G$.
To show that $A\cap B$ is a normal subgroup of $AB$, let $x\in A \cap B$ and $ab\in AB$, where $a\in A$ and $b \in B$.
Then we have the conjugate
\begin{align*}
(ab)x(ab)^{-1}=a(bxb^{-1})a^{-1}. \tag{*}
\end{align*}

We show that the right hand side of (*) is in both $A$ and $B$.
Since $x\in A\cap B \subset A$ and $A$ is a normal subgroup of $G$, we have
\[bxb^{-1}\in A.\]
Thus the right hand side of (*) is in $A$.

Also, since the elements $a, bxb^{-1}, a^{-1}$ are all in the abelian group $A$, we have
\begin{align*}
a(bxb^{-1})a^{-1}=aa^{-1}(bxb^{-1})=bxb^{-1}\in B
\end{align*}
since $x, b\in B$.

Therefore $(ab)x(ab)^{-1}$ is in both $A$ and $B$, and hence
\[(ab)x(ab)^{-1} \in A \cap B\]
and the group $A \cap B$ is a normal subgroup of $AB$.

Two Normal Subgroups Intersecting Trivially Commute Each Other
Let $G$ be a group. Assume that $H$ and $K$ are both normal subgroups of $G$ and $H \cap K=1$. Then for any elements $h \in H$ and $k\in K$, show that $hk=kh$.
Proof.
It suffices to show that $h^{-1}k^{-1}hk \in H \cap K$.
In fact, if this it true then we have […]

If a Sylow Subgroup is Normal in a Normal Subgroup, it is a Normal Subgroup
Let $G$ be a finite group. Suppose that $p$ is a prime number that divides the order of $G$.
Let $N$ be a normal subgroup of $G$ and let $P$ be a $p$-Sylow subgroup of $G$.
Show that if $P$ is normal in $N$, then $P$ is a normal subgroup of $G$.
Hint.
It follows from […]

Two Quotients Groups are Abelian then Intersection Quotient is Abelian
Let $K, N$ be normal subgroups of a group $G$. Suppose that the quotient groups $G/K$ and $G/N$ are both abelian groups.
Then show that the group
\[G/(K \cap N)\]
is also an abelian group.
Hint.
We use the following fact to prove the problem.
Lemma: For a […]

Commutator Subgroup and Abelian Quotient Group
Let $G$ be a group and let $D(G)=[G,G]$ be the commutator subgroup of $G$.
Let $N$ be a subgroup of $G$.
Prove that the subgroup $N$ is normal in $G$ and $G/N$ is an abelian group if and only if $N \supset D(G)$.
Definitions.
Recall that for any $a, b \in G$, the […]

Abelian Group and Direct Product of Its Subgroups
Let $G$ be a finite abelian group of order $mn$, where $m$ and $n$ are relatively prime positive integers.
Then show that there exists unique subgroups $G_1$ of order $m$ and $G_2$ of order $n$ such that $G\cong G_1 \times G_2$.
Hint.
Consider […]

Non-Abelian Simple Group is Equal to its Commutator Subgroup
Let $G$ be a non-abelian simple group. Let $D(G)=[G,G]$ be the commutator subgroup of $G$. Show that $G=D(G)$.
Definitions/Hint.
We first recall relevant definitions.
A group is called simple if its normal subgroups are either the trivial subgroup or the group […]

Non-Abelian Group of Order $pq$ and its Sylow Subgroups
Let $G$ be a non-abelian group of order $pq$, where $p, q$ are prime numbers satisfying $q \equiv 1 \pmod p$.
Prove that a $q$-Sylow subgroup of $G$ is normal and the number of $p$-Sylow subgroups are $q$.
Hint.
Use Sylow's theorem. To review Sylow's theorem, check […]

A Simple Abelian Group if and only if the Order is a Prime Number
Let $G$ be a group. (Do not assume that $G$ is a finite group.)
Prove that $G$ is a simple abelian group if and only if the order of $G$ is a prime number.
Definition.
A group $G$ is called simple if $G$ is a nontrivial group and the only normal subgroups of $G$ is […]