$a \in G$ commutes with all its conjugates iff $a$ belongs to an abelian normal subgroup of $G$
49527
Prove that $a$ commutes with each of its conjugates in $G$ iff $a$ belongs to an abelian normal subgroup of $G$.
3756
If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
151699
Group of inner automorphisms of a group $G$
96878
A finite group with the property that all of its proper subgroups are abelian
51747
Showing that the intersection of all subgroups of order $n$ is normal
41237
$H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
106779
How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
65900
Show that a finite group with certain automorphism is abelian
60208
Proof of property of abelian group
5741
Direct product of abelian groups
25439
f(a) = inverse of a is an isomorphism iff a group 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
97928
Normal subgroup of a normal subgroup
2174
Proving that HK is a subgroup when K is normal
49122
Why commutator subgroup is normal to G?
101360
Why is a normal subgroup of $G_1\times G_2$ with trivial intersections with $G_1$ and $G_2$ is abelian?
70373
Intersection of normal subgroups proof