Prove that $G$ is abelian if and only if the mapping $f : G → G$ defined by $f(g) = g^{-1}$ is a group isomorphism.
25439
f(a) = inverse of a is an isomorphism iff a group G is Abelian
7890
Prove the map $\Phi:G\to\operatorname{Aut}G$ with $g\mapsto (x\mapsto g^{-1}xg)$ is an homomorphism.
134858
Abelian Group Homomorphism
138406
If N and every subgroup of N is normal in G then G/N is abelian .
23579
If $G$ is isomorphic to $H$, then ${\rm Aut}(G)$ is isomorphic to ${\rm Aut}(H)$
68733
If $G/Z(G)$ is cyclic then $G$ is abelian – what is the point?
94090
How to show that for any abelian group $G$, $\text{Hom}(\mathbb{Z},G)$ is isomorphic to $G$
98542
Prove that $H$ is a subgroup of an abelian group $G$
46863
Show that $g$ is injective
46342
Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
85768
The function $f:G \rightarrow G$ defined by $f(x)=x^2$ is a homomorphism iff G is abelian.
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
90332
Prove that that $U(n)$ is an abelian group.
7858
Proving that $f(x)$ is irreducible over $F(b)$ if and only if $g(x)$ is irreducible over $F(a)$
17510
Proving a set is an abelian group.
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
71420
Prove that a group is abelian.