Let $R$ be a finite group which has even number of elements. Prove that at for least two elements $r\in R$ the following equation is valid: $r^{2}=1$.
114590
Prove that any finite group $G$ of even order contains an element of order $2.$
5226
Prove that a finite semigroup G with identity is a group if and only if G contains only one idempotent
84789
Order of products of elements in a finite Abelian group
56650
Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
13153
at least one element fixed by all the group
81543
Finite abelian group contains an element with order equal to the lcm of the orders of its elements
7085
Prove that if the equation $x^{2} \equiv a\pmod{pq}$ has any solutions, then it has four solutions.
8742
Examples and further results about the order of the product of two elements in a group
9110
Let G be a group such that $(xy)^2 = (yx)^2$ for all x, y ∈ G. Show that $xy^2 = y^2x$ for all x, y ∈ G.
10548
The equation $-1 = x^2 + y^2$ in finite fields
65613
$G$ is a finite group and $T$ an automorphism of $G$ such that $T$ sends more than $\frac{3}{4}|G|$ elements to their inverse.
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.
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$
102971
Product of all elements of a finite group with an unique element of order 2
128834
Proving the number of even and odd permutations of a subgroup $H<S_{n}$ are equal, provided $H$ is not contained in $A_{n}$
121423
Prove that every element of a finite group has an order
81864
Finite group of order $mn$ with $\gcd(m,n) = 1$ .