query: Torsion elements of a group aren't necessarily a subgroup
positive_passages.docid: 8742
positive_passages.text: Examples and further results about the order of the product of two elements in a group
negative_passages.docid: 107935
negative_passages.text: If $G$ is a group, $H$ is a subgroup of $G$ and $g\in G$, is it possible that $gHg^{-1} \subset H$?
negative_passages.docid: 150188
negative_passages.text: Construct a non-abelian group of order 75
negative_passages.docid: 151029
negative_passages.text: The number of non normal subgroups of a group cannot be $1$?
negative_passages.docid: 90081
negative_passages.text: problems in group theory
negative_passages.docid: 114662
negative_passages.text: Construction of a specific non-commutative and infinite group (with conditions on the order of the elements)
negative_passages.docid: 39938
negative_passages.text: Subgroups of a cyclic group and their order.
negative_passages.docid: 57883
negative_passages.text: Example of a Subgroup That Is Not Normal
negative_passages.docid: 27780
negative_passages.text: Subgroup which is generated by odd elements
negative_passages.docid: 135241
negative_passages.text: Show that $\mathbb{Z}/n^2\mathbb{Z}$ is not isomorphic to other group
negative_passages.docid: 2254
negative_passages.text: Finite group with isomorphic normal subgroups and non-isomorphic quotients?
negative_passages.docid: 3756
negative_passages.text: If $G$ is abelian, then the set of all $g \in G$ such that $g = g^{-1}$ is a subgroup of $G$
negative_passages.docid: 92298
negative_passages.text: The additive group $\mathbb{G}_a$ over an infinite field $k$ is not a topological group
negative_passages.docid: 93666
negative_passages.text: Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
negative_passages.docid: 34552
negative_passages.text: Proper subgroups of non-cyclic p-group cannot be all cyclic?
negative_passages.docid: 88645
negative_passages.text: Prove that the additive groups $\mathbb{R}$ and $\mathbb{Q}$ are not isomorphic.
negative_passages.docid: 142739
negative_passages.text: Nonempty subset H of group G is subgroup iff $ab^{-1} \in H $ for any $a,b \in H$