Prove that $(a,b)^2={a^2}{b^2}$ for all elements $a,b$ of a group iff the group is an abelian group.
43783
Abelian group (Commutative group)
108310
$(a_1 a_2 \cdots a_n)^2=e$ in a finite abelian group
130327
Prove that $a^2 > b^2$ ; if $a > b \geq 0$.
101543
Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
105832
prove that if $\gcd(a,b)=1$ then $\gcd(a^2, b^2)=1$
109611
If G is a group, prove that $(ab)^n =a^nb^n$ implies G is abelian.
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.
126358
Prove for $\gcd$ of two integers $a,b$, that $(a^n,b^n) = (a,b)^n$
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
81354
The product of all the elements of a finite abelian group
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
65224
Prove that if $a^2 = a$ for all elements of the ring $R$, then $R$ has characteristic 2. Is the converse of the statement true?
27905
Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.
121142
Take $G$ to be a group of order $600$. Prove that for any element a in $G$ there exist a $b$ in $G$ such that $a = b^7$
2771
A group has odd number of elements iff each element is a square, $a=b^2$.
85579
$G$ a group s.t. every non-identity element has order 2. If $G$ is finite, prove $|G| = 2^n$ and $G \simeq C_2 \times C_2 \times\cdots\times C_2$
118467
Prove that $a^2+ab+b^2\ge 0$