Show that in a finite abelian group $G=\{a_1,\dots, a_n\}$, and $x=a_1\dots a_n$, then $x^2=e$
5297
If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
84789
Order of products of elements in a finite Abelian group
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
149750
Help understanding Fundamental Theorem of Finite Abelian Groups
18182
An abelian group $G$ with ${\rm Aut}(G)$ non-abelian
57809
For which $n$, $G$ is abelian?
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
101872
Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
145159
Finite group is abelian if the representatives of its conjugacy classes commute
33593
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
3756
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
11003
Proving that $G/N$ is an abelian group
141125
Group with order $p^2$ must be abelian . How to prove that?
63136
Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic