Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$

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• Prove a Group is Abelian if $(ab)^2=a^2b^2$ Let $G$ be a group. Suppose that \[(ab)^2=a^2b^2\] for any elements $a, b$ in $G$. Prove that $G$ is an abelian group.   Proof. To prove that $G$ is an abelian group, we need \[ab=ba\] for any elements $a, b$ in $G$. By the given […]
• 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 […]
• If Every Nonidentity Element of a Group has Order 2, then it’s an Abelian Group Let $G$ be a group. Suppose that the order of nonidentity element of $G$ is $2$. Then show that $G$ is an abelian group.   Proof. Let $x$ and $y$ be elements of $G$. Then we have \[1=(xy)^2=(xy)(xy).\] Multiplying the equality by $yx$ from the right, we […]
• 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 […]
• 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 […]
• Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian Group Let $A$ be an abelian group and let $T(A)$ denote the set of elements of $A$ that have finite order. (a) Prove that $T(A)$ is a subgroup of $A$. (The subgroup $T(A)$ is called the torsion subgroup of the abelian group $A$ and elements of $T(A)$ are called torsion […]
• Quotient Group of Abelian Group is Abelian Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$. Then prove that the quotient group $G/N$ is also an abelian group.   Proof. Each element of $G/N$ is a coset $aN$ for some $a\in G$. Let $aN, bN$ be arbitrary elements of $G/N$, where $a, b\in […]
• Normal Subgroups Intersecting Trivially Commute in a Group Let $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$. Show that for any $a \in A$ and $b \in B$ we have $ab=ba$. Hint. Consider the commutator of $a$ and $b$, that […]