query: A group of order $p^2$, where $p$ is prime, with exactly one proper subgroup is cyclic.
positive_passages.docid: 19229
positive_passages.text: How to find all groups that have exactly 3 subgroups?
negative_passages.docid: 134820
negative_passages.text: $p$ prime, Group of order $p^n$ is cyclic iff it is an abelian group having a unique subgroup of order $p$
negative_passages.docid: 50762
negative_passages.text: If $S_n$ has cyclic subgroups of these orders or not?
negative_passages.docid: 126893
negative_passages.text: A group of order 2p (p prime) and other conditions - prove abelian.
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: 7973
negative_passages.text: Groups of order $pq$ have a proper normal subgroup
negative_passages.docid: 13734
negative_passages.text: Show that every group of prime order is cyclic
negative_passages.docid: 42389
negative_passages.text: If $(G, \oplus)$ has order $2n$, prove that every proper subgroup of $(G, \oplus)$
negative_passages.docid: 114535
negative_passages.text: what groups have only elements of prime order?
negative_passages.docid: 14528
negative_passages.text: Proof that all abelian simple groups are cyclic groups of prime order
negative_passages.docid: 122335
negative_passages.text: Abelian $p$-group with unique subgroup of index $p$
negative_passages.docid: 141125
negative_passages.text: Group with order $p^2$ must be abelian . How to prove that?
negative_passages.docid: 19668
negative_passages.text: Does every group whose order is a power of a prime p contain an element of order p?
negative_passages.docid: 157841
negative_passages.text: Existence of a non-abelian group of order $p^n$.
negative_passages.docid: 25835
negative_passages.text: Structure of groups of order $pq$, where $p,q$ are distinct primes.
negative_passages.docid: 23885
negative_passages.text: Structure of a group, $G$, of order $pq$ where $p, q$ are prime.
negative_passages.docid: 104460
negative_passages.text: When a group is cyclic