Prove a Group is Abelian if $(ab)^2=a^2b^2$

Abelian Group problems and solutions

Problem 401

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.

 
LoadingAdd to solve later

Sponsored Links

Proof.

To prove that $G$ is an abelian group, we need
\[ab=ba\] for any elements $a, b$ in $G$.

By the given relation, we have
\[(ab)^2=a^2b^2.\] The left hand side is
\[(ab)^2=(ab)(ab),\] and thus the relation becomes
\[(ab)(ab)=a^2b^2.\] Equivalently, we can express it as
\[abab=aabb.\] Multiplying by $a^{-1}$ on the left and $b^{-1}$ on the right, we obtain
\begin{align*}
a^{-1}(abab)b^{-1}=a^{-1}(aabb)b^{-1}.
\end{align*}
Since $a^{-1}a=e, bb^{-1}=e$, where $e$ is the identity element of $G$, we have
\[ebae=eabe.\] Since $e$ is the identity element, it yields that
\[ba=ab\] and this implies that $G$ is an abelian group.

Related Question.

I wondered what happens if I change the number $2$ in $(ab)^2=a^2b^2$ into $3$, and created the following problem:

Problem. If $G$ is a group such that $(ab)^3=a^3b^3$ and $G$ does not have an element of order $3$, then $G$ is an abelian group.

For a proof of this problem, see the post “Prove a group is abelian if $(ab)^3=a^3b^3$ and no elements of order $3$“.


LoadingAdd to solve later

Sponsored Links

More from my site

  • Eckmann–Hilton Argument: Group Operation is a Group HomomorphismEckmann–Hilton Argument: Group Operation is a Group Homomorphism Let $G$ be a group with the identity element $e$ and suppose that we have a group homomorphism $\phi$ from the direct product $G \times G$ to $G$ satisfying \[\phi(e, g)=g \text{ and } \phi(g, e)=g, \tag{*}\] for any $g\in G$. Let $\mu: G\times G \to G$ be a map defined […]
  • If a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian GroupIf a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian Group Let $G$ be a group with identity element $e$. Suppose that for any non identity elements $a, b, c$ of $G$ we have \[abc=cba. \tag{*}\] Then prove that $G$ is an abelian group.   Proof. To show that $G$ is an abelian group we need to show that \[ab=ba\] for any […]
  • Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$ Let $G$ be a group. Suppose that we have \[(ab)^3=a^3b^3\] for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$. Then prove that $G$ is an abelian group.   Proof. Let $a, b$ be arbitrary elements of the group $G$. We want […]
  • If Every Nonidentity Element of a Group has Order 2, then it’s an Abelian GroupIf 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 […]
  • Quotient Group of Abelian Group is AbelianQuotient 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 […]
  • Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian GroupTorsion 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 […]
  • Pullback Group of Two Group Homomorphisms into a GroupPullback Group of Two Group Homomorphisms into a Group Let $G_1, G_1$, and $H$ be groups. Let $f_1: G_1 \to H$ and $f_2: G_2 \to H$ be group homomorphisms. Define the subset $M$ of $G_1 \times G_2$ to be \[M=\{(a_1, a_2) \in G_1\times G_2 \mid f_1(a_1)=f_2(a_2)\}.\] Prove that $M$ is a subgroup of $G_1 \times G_2$.   […]
  • A Group Homomorphism and an Abelian GroupA Group Homomorphism and an Abelian Group Let $G$ be a group. Define a map $f:G \to G$ by sending each element $g \in G$ to its inverse $g^{-1} \in G$. Show that $G$ is an abelian group if and only if the map $f: G\to G$ is a group homomorphism.   Proof. $(\implies)$ If $G$ is an abelian group, then $f$ […]

You may also like...

1 Response

  1. 05/07/2017

    […] (For a proof of this problem, see the post “Prove a group is abelian if $(ab)^2=a^2b^2$“.) […]

Leave a Reply

Your email address will not be published. Required fields are marked *

This site uses Akismet to reduce spam. Learn how your comment data is processed.

More in Group Theory
Group Theory Problems and Solutions in Mathematics
$p$-Group Acting on a Finite Set and the Number of Fixed Points

Let $P$ be a $p$-group acting on a finite set $X$. Let \[ X^P=\{ x \in X \mid g\cdot x=x...

Close