Prove that G is abelian if (G:H) is prime, where H is in Z(G)
119512
If H ≤ Z(G) ≤ G, where G is a finite group,Z(G) is its center, and (G:H) = p for some prime p, then G is abelian.
98544
Prove that in any GCD domain every irreducible element is prime
87882
If $N \lhd G$ is abelian and $G/N$ is abelian 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 $G$.
60208
Proof of property of abelian group
141125
Group with order $p^2$ must be abelian . How to prove that?
97565
Quotient group $G/H$ is abelian iff $[G,G] \subseteq H$
135039
${\rm Aut}(G)$ is cyclic $\implies G$ is abelian
57669
A proof that $\mathbb{Z}_{p}$ is an integral domain if and only if $p$ is prime.
106809
$f: G→G$ defined by $f(x) =x^2$ is a homomorphism if and only if $G$ is abelian.
127603
If $G$ be a group of order $8$ and $x$ be a element of order $4$. Prove that $x^2 \in Z(G)$
344
Prove that if $p\in \mathbb{Z}$ is irreducible, then it is also prime.
42891
Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$
90144
Show that if $a+bi$ is irreducible over $\mathbb{Z} [i]$ then $a^2 + b^2$ is prime over $\mathbb{Z} $
98647
$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian
90332
Prove that that $U(n)$ is an abelian group.
69582
Is it true that: $a+bi$ is prime in $\mathbb{Z}[i]$ if and only if $a^2+b^2$ is prime in $\mathbb{Z}$