The function $f:G \rightarrow G$ defined by $f(x)=x^2$ is a homomorphism iff G is abelian.
46342
Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
90010
Can we ascertain that there exists an epimorphism $G\rightarrow H$?
123419
Abstract algebra: homomorphism and an element of a prime order
55900
Is there a ring homomorphism $M_2(\mathbb Z)\to \mathbb Z$?
67374
Is it true that $\mathbb{R}$ and $\mathbb{R}^2$ are isomorphic as abelian groups?
90811
If $[G' : G'']\leq p^2$, then $G'$ is abelian.
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
78698
Prove if $f$ has a right inverse function: $f\circ g=id_x$ $\iff$ $f$ is onto $Y$
25439
f(a) = inverse of a is an isomorphism iff a group G is Abelian
13764
Homomorphisms from $\mathbb{Z} \rightarrow S_3$.
143451
Homomorphism from $(\Bbb Q,+)$ to a finite group
91947
Proof of $ f:I_n \rightarrow I_m \Rightarrow n = m$
72694
Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$
48537
If $\forall x \in R, x^2-x \in Z(G)$, than $R$ is commutative
22995
Let f:G to H be 1-1 homomorphism show that order of f(a) divides the order of G.
124309
$g\in G$ maximal order in $G$ abelian then $G=\left\oplus H$
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.