Given $(G,*)$ a group. Show the function $f: G \to G$ defined as $f(a)=a^{2}$ is an homomorphism if and only if G is abelian.
46342
Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
32428
Group homomorphisms and which of the following statements are true (NBHM-$2014$)
150961
An example for a homomorphism that is not an automorphism
129452
Groups $G,H$ with surjective homomorphisms that are quotient of each other
33196
show that A(T) is a group under operation of composition of functions
18182
An abelian group $G$ with ${\rm Aut}(G)$ non-abelian
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
51695
Finding all homomorphisms between $\mathbb{Z}/m\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$
94090
How to show that for any abelian group $G$, $\text{Hom}(\mathbb{Z},G)$ is isomorphic to $G$
80601
Let $a,b$ be in a group $G$. Show $(ab)^n=a^nb^n$ $\forall n\in\mathbb{Z}$ if and only if $ab=ba$.
61334
Application of Main Homomorphism Theorem
25439
f(a) = inverse of a is an isomorphism iff a group G is Abelian
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
88930
Homomorphism between free groups
70034
Showing that this set of functions is a group.
120971
(Revised) Prove that $\phi$ is a group homomorphism and find the kernel.