$(g_1 g_2)^m = g_1^m g_2^m \Leftrightarrow g_1 g_2 = g_2 g_1$ -- Question from Mac Lane and Birkhoff (Chapter II, section 3, problem 7)
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
52357
If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
79367
Understanding the difference between two theorems related to $|G\,:\,K|=|H\,:\,K|\cdot|G\,:\,H|$.
110390
Order of group $GL_{2}\left( \mathbb{F}_{p}\right) $
21217
$G$, group. $a,b\in G$. Show that $|a|=|a^{-1}|; |ab|=|ba|$, and $|a|=|cac^{-1}|, \forall c\in G$.
34059
$(a^{2^n}+1,a^{2^m}+1)=1 or 2$
118727
$g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.
51706
If $(k,l) = 1$, show that $(b_2k-b_1l, a_1l-a_2k) = 1$ for $a_1b_2-a_2b_1=1$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
16749
Let $A_i= \left \{...,-2,-1,0,1,...,i \right\}$. Find $\bigcup_{i=1}^{n} A_i$ and $\bigcap_{i=1}^{n} A_i$
124309
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
55976
Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
41731
Question about the order of the product of two elements in an abelian group
102777
Is $G^{(X, Y)} = (G^X)^Y?$ ($A^B$ just means that $B$ is mapped to $A$)
18714
$\gcd\left(a+b,\frac{a^p+b^p}{a+b}\right)=1$, or $p$
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
51858
Showing $\gcd(2^m-1,2^n+1)=1$