Let $a,b\in G$. If $|a|$, $|b|$ finite then $|ab|$ finite for arbitrary group $G$
8742
Examples and further results about the order of the product of two elements in a group
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
59561
Suppose that G is a group and N < G a normal subgroup. Prove G/N is simple
1364
Let $G$ be a group in which $a^2=e$ for all elements of $a$ of $G$. Show that $G$ is Abelian.
118337
Let $S$ be a normal p-subgroup of a finite group $G$. Prove that $S \subseteq P$ for every Sylow p-subgroup $P$ of $G$
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
70394
Prove that $a * b$ is a group (Hungerford, 2nd ed., sec $7.2$, Exercise $22$)
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
83938
If$(ab)^n=a^nb^n$ & $(|G|, n(n-1))=1$ then $G$ is abelian
29996
Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
58654
If $|A| > \frac{|G|}{2} $ then $AA = G $
103216
The set $\{g^2 | g \in G\}$ in a group $G$
145159
Finite group is abelian if the representatives of its conjugacy classes commute
156429
If $G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.
52121
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
15633
If G is a group of even order, prove it has an element $a \neq e$ satisfying $a^2 = e$.