$G$ is a group ִ$a\in G$ we know that $O(a)=n$ then $O(a)=O(a^{-1})$
5089
An element of a group has the same order as its inverse
112157
If $g$ is the generator of a group $G$, order $n$, when is $g^k$ a generator?
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
47179
Prove that $[N, G']=1$ when $N$ is a nilpotent subgroup of a group $G$?
117468
$o(g^{k})=\frac{o(g)}{\gcd(o(g),k)}$ order of a power of a group element.
151699
Group of inner automorphisms of a group $G$
107935
If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
108310
$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group
71420
Prove that a group is abelian.
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 $x^2=e$.
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
57809
For which $n$, $G$ is abelian?
95378
Known bounds for the number of groups of a given order.
30676
$G'$ and the product of all the elements in an odd order group
115204
Show that SO(n) is a normal subgroup of O(n)
101596
Determine group G as cyclic
79452
How to prove that it is a group?