If $G$ is a non-abelian group of order $p^{3}$ for prime p , then the center of $G$ is generated by all elements of the form $aba^{-1}b^{-1}.$
43580
Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
92493
If $G$ is non-abelian, then $Inn(G)$ is not a normal subgroup of the group of all bijective mappings $G \to G$
55976
Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
138410
Centralizer of $\mathbb{C}[G]$ in $\mathbb{C}[H]$
29802
Part of simple proof of nontrivial center in p-group
61546
Prove that for each prime $p$ there exists a nonabelian group of order $p^3$
11003
Proving that $G/N$ is an abelian group
33728
Basis for Center of Group Ring
85011
Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
27905
Let $G$ be a group, where $(ab)^3=a^3b^3$ and $(ab)^5=a^5b^5$. Prove that $G $ is an abelian group.
88718
A group is generated by two elements of order $2$ is infinite and non-abelian
114320
Showing that if the center of a group is trivial, then the center of its automorphism group is trivial
133627
If $\phi(g)=g^3$ is a homomorphism and $3 \nmid |G|$, $G$ is abelian.
106577
If G is not abelian with order $p^3$ then $G'=Z(G)$
108830
Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
122970
A question on $p$-groups, and order of its commutator subgroup.
134820
$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$