query: Prove that a finite $p-$group $G$ with a unique subgroup $H$ of index $p$ is cyclic.
positive_passages.docid: 122335
positive_passages.text: Abelian $p$-group with unique subgroup of index $p$
negative_passages.docid: 92430
negative_passages.text: Show that the group is cyclic.
negative_passages.docid: 126590
negative_passages.text: Why is the multiplicative group of a finite field cyclic?
negative_passages.docid: 146760
negative_passages.text: Prove that H is a subgroup of G
negative_passages.docid: 152662
negative_passages.text: Proving that any finite cyclic group of order $n$ is isomorphic to $$
negative_passages.docid: 107716
negative_passages.text: Show that $G$ is cyclic
negative_passages.docid: 45727
negative_passages.text: If $G$ contains a normal subgroup $H \cong \mathbb{Z_2}$ such that $G/H$ is infinite cyclic, then $G \cong \mathbb{Z} \times \mathbb{Z_2}$
negative_passages.docid: 2104
negative_passages.text: Finite group for which $|\{x:x^m=e\}|\leq m$ for all $m$ is cyclic.
negative_passages.docid: 106277
negative_passages.text: Prove that $G$ contains a subgroup $H_i$ for each $i\in\Bbb Z$ with $\dots\subset H_{−2}\subset H_{−1}\subset H_0\subset H_1\subset\dots?$
negative_passages.docid: 61513
negative_passages.text: Prove that all cyclic groups of the same order are isomorphic
negative_passages.docid: 85011
negative_passages.text: Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
negative_passages.docid: 106660
negative_passages.text: Every subgroup of the integers has finite index
negative_passages.docid: 68201
negative_passages.text: Cyclic group of order $n$
negative_passages.docid: 151619
negative_passages.text: A finite abelian group containing a non-trivial subgroup which lies in every non-trivial subgroup is cyclic
negative_passages.docid: 101596
negative_passages.text: Determine group G as cyclic
negative_passages.docid: 150042
negative_passages.text: Prove $x^{-1}Hx$ is a subgroup of $G$
negative_passages.docid: 69046
negative_passages.text: Proof of $\mathbb{Q}$ is not cyclic