Let $G$ a finite group with order of $2p$, where $p > 2$ is prime. given that there's $a \in Z(G)$ such that $o(a) = 2$. Prove: $G$ is abelian.
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 $D_p$
47239
For $p,q$ primes such that $q|p-1$, show there is only one non-abelian group of order $pq$, up to isomorphism
59811
Finite Abelian groups with the same number of elements for all orders are isomorphic
51373
$G$ non abelian, order $p^3$ ($p$ prime). Suppose that the center is $p^2$, prove that $\exists\ x$ outside of the center, of order p
7121
If $G$ is a finite group and $|G| < |A| + |B|$, then $G=AB$.
14528
Proof that all abelian simple groups are cyclic groups of prime order
134376
Let $G$ be a nonabelian group. Let $a \in G$ have order $4$ and let $b \in G$ have order $3$. Then the order of the element $ab$ in $G$
145127
For any integers $m,n>1$ , does there exist a group $G$ with elements $a,b \in G$ such that $o(a)=m , o(b)=n$ but $ab$ has infinite order ?
81543
Finite abelian group contains an element with order equal to the lcm of the orders of its elements
48585
Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
125559
In a group $G$, if for all $a,b,c\in G$, $ab=ca\Rightarrow b=c$, then $G$ is abelian
106577
If G is not abelian with order $p^3$ then $G'=Z(G)$
134820
$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
70001
Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)
57809
For which $n$, $G$ is abelian?
13631
Showing group with $p^2$ elements is Abelian
65900
Show that a finite group with certain automorphism is abelian