Let $G$ be a group. Prove that $o(xy) = o(yx)$ for all $x,y \in G$
98734
Order of a product of elements in a generic group
124309
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
114522
Prove that identity is unique in a group
63136
Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic
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 $x^2=e$.
92057
Prove that for every odd integer n, there is a group...
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
42891
Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$
115204
Show that SO(n) is a normal subgroup of O(n)
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
150199
How to prove for a finite group that $|\langle g \rangle|=o(g)$?
81354
The product of all the elements of a finite abelian group
79452
How to prove that it is a group?
105910
How to prove that $[G:xHx^{-1}] = [G:H]$ given $H \le G$?
103216
The set $\{g^2 | g \in G\}$ in a group $G$
73571
Proof of Cauchy's Theorem for abelian groups