Prove $G$ is cyclic if for every $n\in \Bbb{N}$, there are at most $n$ solutions to the equation $x^n=e$.
35664
If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic
20679
Prove that if $g^2=e$ for all $g$ in $G$ then $G$ is Abelian.
59533
If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic.
11003
Proving that $G/N$ is an abelian group
142894
Showing that a group of order $pq$ is cyclic if it has normal subgroups of order $p$ and $q$
48081
Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
61394
The product of finitely many cyclic groups is cyclic
104855
Prove that $N \subset Z(G)$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
56464
Proving the irrationality of $e^n$
117352
Show $x^n+ax+b=0$ has most two solutions
46355
If $[G:H]=n$, is it true that $x^n\in H$ for all $x\in G$?
147574
Prove of a finite cyclic group of order $n$ is isomorphic tto $(\mathbb Z_n,+)$.
144258
How to show a group is cyclic?
44729
Prove $g$ is a generator if $g^q=1 \pmod p$
2043
$X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$
114379
Show that number of solutions satisfying $x^5=e$ is a multiple of 4?