Suppose G is a finite abelian group with a nontrivial subgroup H contained in every subgroup of G. Show that G is cyclic.
134820
$p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
94239
Proof that if group $G/Z(G)$ is cyclic, then $G$ is commutative
51359
Prove that every subgroup of an abelian group is a normal subgroup.
17077
Prove or disprove: If $H$ and $G/H$ are cyclic, then $G$ is cyclic.
87882
If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
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.
126590
Why is the multiplicative group of a finite field cyclic?
154398
Show that H is a subgroup of G
45727
If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
71420
Prove that a group is abelian.
152662
Proving that any finite cyclic group of order $n$ is isomorphic to $$
145159
Finite group is abelian if the representatives of its conjugacy classes commute
14255
Commutator subgroup and cyclic normal subgroup
117899
At most one subgroup of every order dividing $\lvert G\rvert$ implies $G$ cyclic
51352
Subgroups of a finite abelian group
76317
Let $C$ be the commutator subgroup of $G$. Prove that $G/C$ is abelian
151025
If G is a group of order n=35, then it is cyclic