Proving or disproving that a nonidentity cyclic group has at least two generators.
58397
Every cyclic group with order > 2 has at least 2 distinct generators
87534
Finitely generated group which is not finitely presented
114522
Prove that identity is unique in a group
107716
Show that $G$ is cyclic
144258
How to show a group is cyclic?
122581
Prove or disprove that $G_1/H_1 \cong G_2/H_2$
59902
Example of $2$ non-isomorphic groups that have the same quotients
77128
Necessary and Sufficient conditions for $(i \, j)$ and $(1 \, 2 \, \dotsc \, n)$ generate $S_n$.
126927
$\mathbb Q$ is not the direct product of two non-trivial groups.
73248
Prove that $\mathbb{R^*}$, the set of all real numbers except $0$, is not a cyclic group
125668
A proof that BS(1,2) is not polycyclic
33593
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$
36040
A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.
70688
How can I prove that the additive group of rationals is not isomorphic to a direct product of two nontrivial groups?
117178
Cyclic group generators
135710
Cyclic Automorphism group
14139
$(12)$ and $(123\dots n)$ are generators of $S_n$