Let G be a group. Prove that the function: $G→G, x→x^2$ is a group homomorphism if and only if G is an abelian group.
106809
$f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
82935
Prove that f∈Θ(g)if and only ifO(f) =O(g).
159005
Finding all homomorphisms between two groups - couple of questions
13631
Showing group with $p^2$ elements is Abelian
77221
Prove that $G$ has a subgroup isomorphic to $G/H$.
4883
Let $H$ be a subgroup of a group $G$ such that $x^2 \in H$ , $\forall x\in G$ . Prove that $H$ is a normal subgroup of $G$
150042
Prove $x^{-1}Hx$ is a subgroup of $G$
40993
Let $f:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic.
101543
Let $A$ be a group, where $a^2=1$, a belongs to $A$. Prove that this group is commutative.
64419
How to construct a homomorphism from an abelian group to $\mathbb{Q}/\mathbb{Z}$?
154398
Show that H is a subgroup of G
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 $G$?
94090
How to show that for any abelian group $G$, $\text{Hom}(\mathbb{Z},G)$ is isomorphic to $G$
75473
If a group $G$ has odd order, then the square function is injective.
5297
If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.
90811
If $[G' : G'']\leq p^2$, then $G'$ is abelian.
83710
Order of a homomorphism of groups