If Every Nonidentity Element of a Group has Order 2, then it’s an Abelian Group

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• 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 […]
• Order of the Product of Two Elements in an Abelian Group Let $G$ be an abelian group with the identity element $1$. Let $a, b$ be elements of $G$ with order $m$ and $n$, respectively. If $m$ and $n$ are relatively prime, then show that the order of the element $ab$ is $mn$.   Proof. Let $r$ be the order of the element […]
• 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 […]
• 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 […]
• Order of Product of Two Elements in a Group Let $G$ be a group. Let $a$ and $b$ be elements of $G$. If the order of $a, b$ are $m, n$ respectively, then is it true that the order of the product $ab$ divides $mn$? If so give a proof. If not, give a counterexample.   Proof. We claim that it is not true. As a […]
• Elements of Finite Order of an Abelian Group form a Subgroup Let $G$ be an abelian group and let $H$ be the subset of $G$ consisting of all elements of $G$ of finite order. That is, \[H=\{ a\in G \mid \text{the order of $a$ is finite}\}.\] Prove that $H$ is a subgroup of $G$.   Proof. Note that the identity element $e$ of […]
• If a Group is of Odd Order, then Any Nonidentity Element is Not Conjugate to its Inverse Let $G$ be a finite group of odd order. Assume that $x \in G$ is not the identity element. Show that $x$ is not conjugate to $x^{-1}$.   Proof. Assume the contrary, that is, assume that there exists $g \in G$ such that $gx^{-1}g^{-1}=x$. Then we have \[xg=gx^{-1}. […]
• Finite Group and a Unique Solution of an Equation Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that \[b^m=a.\]   We give two proofs. Proof 1. Since $m$ and $n$ are relatively prime […]