If $G$ is a group and $a,\,b\in G$ s.t. $a$ has order $5$ and $a^3b=ba^3$, prove $ab=ba$
102742
order $a$ = 5, $a^3b = ba^3$. show that that $ab = ba$.
156909
$\Bbb{Z}_{26} \rtimes_{\alpha} \Bbb{Z}_{5} \cong \Bbb{Z}_{26} \times \Bbb{Z}_{5}$ as groups.
56607
gcd and order of elements of group
44309
Prove $5a^2 ≠ 3b^2$ for all non-zero rational numbers $a$ and $b$
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.
45760
if G is generated by {a,b} and ab=ba, then prove G is Abelian
110256
Let $a,b \in {\mathbb{Z_+}}$ such that $a|b^2, b^3|a^4, a^5|b^6, b^7|a^8 \cdots$, Prove $a=b$
157582
Inverse of $ab$ and its order in a group with relations $a^2=e=b^6$ and $ab=b^4a$
74586
Prove that every group of order $4$ is abelian
43728
Proof via Group Theory : $\mathrm{lcm}(a,b) \cdot \gcd(a,b) = |ab|$
85106
Group of order 10 has an element of order 5, without using Cauchy's or Sylow's theorems
11003
Proving that $G/N$ is an abelian group
102859
Proving that $G$ is a group if $a*x=b$ and $y*a=b$ have solutions.
85508
Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
30676
$G'$ and the product of all the elements in an odd order group
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
37061
The order of $ab$ when $a,b$ commute