If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
26398
Group of even order contains an element of order 2
158308
Proving that if $a,b$ are even, then $\gcd(a,b) = 2 \gcd(a/2, b/2)$
79452
How to prove that it is a group?
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
81116
$gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
100605
let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
158490
Prove that $G^{op} \cong G$ when $G^{op}$ is the opposite group of G
17047
Prove that if n is even then $n^2$ is even and if n is odd then $n^2$ is odd?
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
1577
Order of nontrivial elements is 2 implies Abelian group
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 $n$
108310
$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group
61550
Existence of elements of order $p^k$ in a finite group
145127
For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
2794
Prove that if a group contains exactly one element of order 2, then that element is in the center of the group.
56607
gcd and order of elements of group