query: Let $G$ be a group, $H$ be a subgroup of $G$ and $h$ be a fixed element of $G$.
positive_passages.docid: 150042
positive_passages.text: Prove $x^{-1}Hx$ is a subgroup of $G$
negative_passages.docid: 13059
negative_passages.text: If a group is the union of two subgroups, is one subgroup the group itself?
negative_passages.docid: 2794
negative_passages.text: Prove that if a group contains exactly one element of order 2, then that element is in the center of the group.
negative_passages.docid: 106779
negative_passages.text: How to show if $H,K$ are normal subgroups in a group $G$ then $[H,K] \leq H \cap K$
negative_passages.docid: 141043
negative_passages.text: $H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
negative_passages.docid: 80273
negative_passages.text: Prove the centralizer of an element in group $G$ is a subgroup of $G$
negative_passages.docid: 64269
negative_passages.text: Finding the minimal $n$ such that a given finite group $G$ is a subgroup of $S_n$
negative_passages.docid: 124709
negative_passages.text: Suppose G be a non-abelian group and H,K be two abelian subgroups of G. Then must HK be an abelian subgroup of G?
negative_passages.docid: 66510
negative_passages.text: A group is abelian if and only if the center of the group is all the group
negative_passages.docid: 91021
negative_passages.text: Torsion elements of a group aren't necessarily a subgroup
negative_passages.docid: 10602
negative_passages.text: Monotone Union of subgroups being subgroup
negative_passages.docid: 111886
negative_passages.text: Let $(G,•)$ be a group in which $(a•b)^3 = a^3•b^3$. Prove that $H=\{x^3 : x \in G\}$ is a normal subgroup of $G$.
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: 104080
negative_passages.text: Group with order $pq$ has subgroups of order $p$ and $q$
negative_passages.docid: 114221
negative_passages.text: A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$
negative_passages.docid: 116080
negative_passages.text: Let G be a finite group with more than one element. Show that G has an element of prime order
negative_passages.docid: 5297
negative_passages.text: If G is a finite abelian group and $a_1,...,a_n$ are all its elements, show that $x=a_1a_2a_3...a_n$must satisfy $x^2=e$.