Let $a,b \in$ group $G$ such that $ab^3a^{-1}=b^2, b^{-1}a^2b=a^3$ Prove that $a=b=e$ (identity)
8483
A Particular Two-Variable System in a Group
58654
If $|A| > \frac{|G|}{2} $ then $AA = G $
108310
$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group
142346
The conditions $a^2b^2=b^2a^2$ and $a^3b^3=b^3a^3$ make the group $G$ abelian?
52357
If $a,b \in G$, show that $ab^{n}a^{-1}=(aba^{-1})^{n}$.
34693
Prove $E^{\mathbb Q}[Y]=E^{\mathbb P}[XY]$ if $E^{\mathbb P}[X]=\mathbb P(X>0)=1$ and $ \mathbb Q(A)=E^{\mathbb Q}[X1_A] $
158490
Prove that $G^{op} \cong G$ when $G^{op}$ is the opposite group of G
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.
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
41977
If $G$ is a group, show that $x^2ax=a^{-1}$ has a solution if and only if $a$ is a cube in $G$
33593
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
92873
To prove in a Group Left identity and left inverse implies right identity and right inverse
125582
Let $\phi:G_1\to G_2$ be a group homomorphism. Show $\phi(g^{-1})=(\phi(g))^{-1}$.
6153
Suppose that $G$ is a group with the property that for every choice of elements in $G$, $axb=cxd$ implies $ab=cd$. Prove that $G$ is Abelian.
132098
Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$.
6012
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
105832
prove that if $\gcd(a,b)=1$ then $\gcd(a^2, b^2)=1$