query: If every subgroup of $G$ is $G^k$ for some k, then $G$ is a cyclic group.
positive_passages.docid: 152071
positive_passages.text: Group $G$ is cyclic $\iff$ every subgroup of $G$ has the form $G^k$, where $G^k=\{g^k\mid g\in G\}$
negative_passages.docid: 55976
negative_passages.text: Are groups satisfying $g=g^{-1}$ for all $g \in G$ abelian?
negative_passages.docid: 89382
negative_passages.text: In a group $G$ where $\exists!$ nontrivial, proper subgroup, show that $G$ is cyclic and $\lvert G\rvert=p^2$, for $p$ prime.
negative_passages.docid: 41237
negative_passages.text: $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
negative_passages.docid: 23394
negative_passages.text: If $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 48585
negative_passages.text: Proving that if $G/Z(G)$ is cyclic, then $G$ is abelian
negative_passages.docid: 61379
negative_passages.text: Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
negative_passages.docid: 40993
negative_passages.text: Let $f:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic.
negative_passages.docid: 84823
negative_passages.text: When is the group of units in $\mathbb{Z}_n$ cyclic?
negative_passages.docid: 75986
negative_passages.text: If all Subgroups are Cyclic, is group Cylic?
negative_passages.docid: 44152
negative_passages.text: If $G$ has only one subgroup of order $n$, then that Subgroup is Normal
negative_passages.docid: 85508
negative_passages.text: Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup.
negative_passages.docid: 57887
negative_passages.text: When are the product of cyclic groups also cyclic?
negative_passages.docid: 101596
negative_passages.text: Determine group G as cyclic
negative_passages.docid: 98739
negative_passages.text: If $H$ is a subgroup of $G$ and $K$ is a subgroup of $H$, then $$|G:K|=|G:H||H:K|$$
negative_passages.docid: 5223
negative_passages.text: $\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$
negative_passages.docid: 154398
negative_passages.text: Show that H is a subgroup of G