Let $G$ be a finite abelian group with exactly one element of order $2$ denoted by $\alpha$. Prove that $\prod_{g \in G} g = \alpha$.
63359
Show that the product of all group elements is $e$ or $u$ if $G$ has exactly one involution $u$
16476
Order of the product of two commuting elements with coprime orders in a group.
1577
Order of nontrivial elements is 2 implies Abelian group
2731
Groups of even order have an element whose order is 2.
143550
Order of product of two elements in an abelian group
65900
Show that a finite group with certain automorphism is abelian
141125
Group with order $p^2$ must be abelian . How to prove that?
145406
Prove every element of $G$ has finite order.
49768
If Gal(K,Q) is abelian then |Gal(K,Q)|=n
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
68971
Elementary approach to proving that a group of order 9 is Abelian
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
7079
ord $(b)|\max\{\text{ord}(g)|g\in G\}$ for all $b\in G\,$ a finite abelian group
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 $x^2=e$.
46687
Elements of $GL_{2}(\mathbb{Z})$ of finite order
59811
Finite Abelian groups with the same number of elements for all orders are isomorphic
138862
Product of all elements in an odd finite abelian group is 1