$G\cong H$. $G$ is cyclic if and only if $H$ is cyclic.
110808
Isomorphisms preserve cyclic
84501
Proof involving Cyclic group, generator and GCD
105951
Does $G\times K\cong H\times K$ imply $G\cong H$?
35800
If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.
62000
$G$ is characteristically simple $\iff$ there is simple $T$ such that $G \cong T\times T \times \cdots \times T$
98739
If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
158074
Cyclic group $G$ with generator $ab$.
155219
Prove or Disprove: If every nontrivial subgroup of a group $G$ is cyclic, then $G$ is cyclic.
84823
When is the group of units in $\mathbb{Z}_n$ cyclic?
35188
Counterexample: $G \times K \cong H \times K \implies G \cong H$
142375
Prove or disprove: If H is a normal subgroup of G such that H and G/H are abelian, then G is abelian.
111703
If $H \leq G$ and $[G:H]! \leq |G|$ then $G$ is not simple
146760
Prove that H is a subgroup of G
36887
Suppose $G$ is a group of order 4. Show either $G$ is cyclic or $x^2=e$.
144258
How to show a group is cyclic?
58014
Prove or disprove: The automorphism group of a finite cyclic group must be cyclic.
40993
Let $f:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic.