Finding $O(ab)$ when $O(a)$ and $O(b)$ are given in non abelian Group
134376
Let $G$ be a nonabelian group. Let $a \in G$ have order $4$ and let $b \in G$ have order $3$. Then the order of the element $ab$ in $G$
67554
In a group, the number of non-self-inverse elements is even
4838
For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
63136
Prove $G$ is Abelian if $N$ is in the centre of $G$ and $G/N$ is cyclic
11003
Proving that $G/N$ is an abelian group
27905
Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.
15341
Non-abelian finite group in which more than half of the elements have order $2$
150585
Group with $a=a^{-1}$ for all $a\in G$ is abelian
7079
ord $(b)|\max\{\text{ord}(g)|g\in G\}$ for all $b\in G\,$ a finite abelian group
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
29717
Not sure how to go about this group theory proof
75590
How many normal subgroups does a non-abelian group $G$ of order $ 21$
143550
Order of product of two elements in an abelian group
83938
If$(ab)^n=a^nb^n$ & $(|G|, n(n-1))=1$ then $G$ is abelian
43783
Abelian group (Commutative group)
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
27703
an example of a non-abelian group $G$ containing a proper normal subgroup $N$ such that $G/N$ is abelian.