Suppose that G is a finite group of even order. Show that G must contain an element $a\ne e$ for which $a^2 = e$
26398
Group of even order contains an element of order 2
96123
Element of maximal order in alternating group $A_{10}$: how to prove this is it?
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
98734
Order of a product of elements in a generic group
114221
A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$
121423
Prove that every element of a finite group has an order
65900
Show that a finite group with certain automorphism is abelian
141125
Group with order $p^2$ must be abelian . How to prove that?
157806
Order of a element on a Symmetric Group
91774
Existence of group $G$ with $|G|>2$ and automorphism group of odd order
36147
A finite group is nilpotent iff two elements with relatively prime order commute
101345
Suppose $n$ is an even positive integer and $H$ is a subgroup of $\mathbb Z/n \mathbb Z$. Prove that either every element of $H$ is...
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
103423
Normal subgroup where G has order prime
81354
The product of all the elements of a finite abelian group
121077
Product of all elements in finite group