query: There are exactly two abelian groups of order $p^2$
positive_passages.docid: 149750
positive_passages.text: Help understanding Fundamental Theorem of Finite Abelian Groups
negative_passages.docid: 84789
negative_passages.text: Order of products of elements in a finite Abelian group
negative_passages.docid: 18415
negative_passages.text: Direct sum of Abelian groups and Isomorphism
negative_passages.docid: 933
negative_passages.text: How many groups exist with order $n$ (two isomorphic groups are treated as the same group)
negative_passages.docid: 108830
negative_passages.text: Let $G$ be a group of order $2p$ , where $p$ is a prime greater than $2$. Then, G is isomorphic to $\mathbb{Z}_{2p}$ or $D_p$
negative_passages.docid: 5741
negative_passages.text: Direct product of abelian groups
negative_passages.docid: 108425
negative_passages.text: Problem on abelian group
negative_passages.docid: 46260
negative_passages.text: Prove that a group generated by two elements of order $2$, $x$ and $y$, is isomorphic to $D_{2n}$, where $n = |xy|.$
negative_passages.docid: 70001
negative_passages.text: Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)
negative_passages.docid: 152481
negative_passages.text: If order of group is $p^2$, where $p$ is prime, how can you deduce $G$ is isomorphic to $C_{p^2}$ or $C_p \times C_p$?
negative_passages.docid: 114590
negative_passages.text: Prove that any finite group $G$ of even order contains an element of order $2.$
negative_passages.docid: 58925
negative_passages.text: Order of general linear group of $2 \times 2$ matrices over $\mathbb{Z}_3$
negative_passages.docid: 51196
negative_passages.text: A group of order $p^2q^2$ is never simple
negative_passages.docid: 143314
negative_passages.text: Sylow subgroup of $S_{p^2}$
negative_passages.docid: 26398
negative_passages.text: Group of even order contains an element of order 2
negative_passages.docid: 8602
negative_passages.text: Galois Group of $(x^2-p_1)\cdots(x^2-p_n)$
negative_passages.docid: 71420
negative_passages.text: Prove that a group is abelian.