Let $G$ be a group. Define $f:G\to G, a↦a^2$ for all $a∈G$. Prove that $f$ is a group homomorphism if and only if $G$ is abelian.
46342
Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
86823
proof verification: If $f:G \rightarrow H$ is group homomorphism, and $H$ is abelian, then $G$ is abelian
133627
If $\phi(g)=g^3$ is a homomorphism and $3 \nmid |G|$, $G$ is abelian.
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
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.
79933
Let $G$ be a group and $c\in G$. Define $*$ on $G$ by: $x*y=xdy$ for all $x,y\in G$, where $d$ is the inverse of $c$. Prove $(G,*)$ is a group.
112993
Is every quotient of a finite abelian group $G$ isomorphic to some subgroup of $G$?
82935
Prove that f∈Θ(g)if and only ifO(f) =O(g).
43759
Prove that the set $\mathrm{Aut}(G)$ of all automorphisms of the group $G$ with the operation of taking the composition is a group
90616
When is mapping $g$ to $g^{-1}$ a group homomorphism?
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
85768
The function $f:G \rightarrow G$ defined by $f(x)=x^2$ is a homomorphism iff G is abelian.
82037
Suppose $G$ is a group generated by elements $x$ and $y$ where $xy^2 = y^3x$ and $yx^3 = x^2y$ What can you prove about $G$?
141125
Group with order $p^2$ must be abelian . How to prove that?
120971
(Revised) Prove that $\phi$ is a group homomorphism and find the kernel.
126930
Finding all group homomorphisms of $S_n\to\mathbb{C}^*$ and $A_n\to\mathbb{C}^*$
140334
Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.