If G is a finite group of even order then G contains an element $a\neq e$ such that $a^2=e$.
26398
Group of even order contains an element of order 2
158308
Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$
71608
A property of finite field of order $2^n$
13355
Product of elements of a finite abelian group
30676
$G'$ and the product of all the elements in an odd order 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 $x^2=e$.
145563
In an abelian group, the elements of finite order form a subgroup.
158282
Finite order of an element $f(g)$ of a homomorphism $f$
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
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 $n$
2794
Prove that if a group contains exactly one element of order 2, then that element is in the center of the group.
152604
For a finite group of order $2n$ does there exist $x$ such that $x\ast x=e$?
90811
If $[G' : G'']\leq p^2$, then $G'$ is abelian.
58654
If $|A| > \frac{|G|}{2} $ then $AA = G $
28676
Is a finite group uniquely determined by the orders of its elements?
12455
$|G|>2$ implies $G$ has non trivial automorphism
13834
Show that G is a group, if G is finite, the operation is associative, and cancellation law holds