Let G be non abelian and |G| = $p^3$ , where p is a prime. Prove that Z(G) = G′ . What is the order of Z(G)?
43580
Let $p$ be a prime number and $G$ a non abelian group or order $p^3$. Prove that $Z(G) = [G,G]$.
121481
Non-abelian finite $p$-group in which every element has prime order?
125640
Group of order $p^2$ is abelian.
22301
If $G$ has no proper subgroup, then $G$ is cyclic of prime order
47239
For $p,q$ primes such that $q|p-1$, show there is only one non-abelian group of order $pq$, up to isomorphism
17178
Order of $\phi(g)$ divides the order of $g$
23394
If $G/Z(G)$ is cyclic, then $G$ is abelian
100605
let $G$ to be finite abelian group of order O(G), let n to be prime number and (O(G),n)=1 prove that $g=x^n$ for any $ x \in G$
111886
Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
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.
156429
If $G$ is a group whereby $(a\cdot b)^{i} =a^i\cdot b^i$ for three consecutive integers $i$ for all $a, b \in G$, show $G$ is abelian.
116080
Let G be a finite group with more than one element. Show that G has an element of prime order
45760
if G is generated by {a,b} and ab=ba, then prove G is Abelian
141890
$G$ is a Group of order $pq$, if $G$ has exactly one subgroup of order $p$ and another with order $q$, then $G$ is cyclic ($p,q$ are prime)
56607
gcd and order of elements of group
133627
If $\phi(g)=g^3$ is a homomorphism and $3 \nmid |G|$, $G$ is abelian.
15387
Order of $GL(n, \mathbb Z_p)$