$G$ satisfying $(ab)^n=a^nb^n$
90081
problems in group theory
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
142346
The conditions $a^2b^2=b^2a^2$ and $a^3b^3=b^3a^3$ make the group $G$ abelian?
150958
Group theory question. If $m$ and $n$ are relatively prime, and $a$, $b$ belong to group and $a^m = b^m$ and $a^n = b^n$, how would one prove $a = b$?
23590
Show that $\gcd\left(\frac{a^n-b^n}{a-b},a-b\right)=\gcd(n d^{n-1},a-b)$
15626
Relations of a~b iff b = ak^2
78298
On the nilpotence of the matrix $AB-BA$
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
144732
If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
52357
If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
120642
Let $g : \Bbb N \times \Bbb N \to\Bbb N \times \Bbb N$ defined as $g(m,n) = (m + n,m - n)$
127293
Prove that $(a-b) \mid (a^n-b^n)$
7015
Prove that if $(ab)^i = a^ib^i \forall a,b\in G$ for three consecutive integers $i$ then G is abelian
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
126280
In a P.I.D., if $a^m = b^m$ and $a^n = b^n$ for $m, n \in \mathbb{N}$ with $\gcd(m,n) = 1$, then $a=b$
46780
Proving $((a\bmod n) (b\bmod n))\bmod n = ab\bmod n $
49768
If Gal(K,Q) is abelian then |Gal(K,Q)|=n