Show that if $G$ is a finite group of even order, there must exist an element $a\in G, a\ne e$ s.t. $a^2=e$
26398
Group of even order contains an element of order 2
10403
Let $p$ be a prime integer. Show that for each $a ∈ GF(p)$ there exist elements $b$ and $c$ of $GF(p)$ satisfying $a = b^2 + c^2$.
13355
Product of elements of a finite abelian group
43059
Show that $g^k=e$ if and only if the order of $g$ is finite and divides $k$.
1577
Order of nontrivial elements is 2 implies Abelian group
2731
Groups of even order have an element whose order is 2.
28676
Is a finite group uniquely determined by the orders of its elements?
34567
If $G_1$ and $G_2$ are finite group and $H\leq G_1\times G_2$ then $H=P_1\times P_2$ with $P_i\leq G_i$?
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
2771
A group has odd number of elements iff each element is a square, $a=b^2$.
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$
145406
Prove every element of $G$ has finite order.
152741
Cardinal of a group $G$ such that for all $x\in G$ we have $x^2=e$
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 ?
13834
Show that G is a group, if G is finite, the operation is associative, and cancellation law holds
56607
gcd and order of elements of group
5089
An element of a group has the same order as its inverse