If $ G $ has no non-trivial automorphism, then $ G $ is abelian and $ g^2 = e $ for all $ g \in G $ .
12455
$|G|>2$ implies $G$ has non trivial automorphism
101079
Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$
106809
$f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
108425
Problem on abelian group
57464
If $G$ is a group show that if $(a \cdot b)^2 = a^2 \cdot b^2$ then $G$ must be abelian.
55976
Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
108213
$N_1,N_2,N_3 \unlhd G, N_i\cap N_j =\{e\}, G = N_iN_j$. Want to show that $G$ is abelian, $N_i$ are isomorphic.
116125
A group $G$ with $G'$ abelian and every abelian normal subgroup finite
133664
When the automorphism group is trivial......
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 $G$?
11113
Group With an Endomorphism That is "Almost" Abelian is Abelian.
9110
Let G be a group such that $(xy)^2 = (yx)^2$ for all x, y ∈ G. Show that $xy^2 = y^2x$ for all x, y ∈ G.
51145
Is it possible to prove $g^{|G|}=e$ in all finite groups without talking about cosets?
36887
Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
88718
A group is generated by two elements of order $2$ is infinite and non-abelian
43580
Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.