query: $(H,\circ)$ is a subgroup of $(G,\circ)$
positive_passages.docid: 142228
positive_passages.text: Proof that a subset closed under group operation of a finite group is a subgroup
negative_passages.docid: 103216
negative_passages.text: The set $\{g^2 | g \in G\}$ in a group $G$
negative_passages.docid: 36709
negative_passages.text: subgroup of direct product of two groups
negative_passages.docid: 11957
negative_passages.text: G is the product of its Sylow subgroups
negative_passages.docid: 140334
negative_passages.text: Let $G$ be a group and $H$ be a subgroup of $G$ of index $2$ .Then show that if $a,b\notin H$ then $ab\in H$.
negative_passages.docid: 37971
negative_passages.text: Prove that the center of a group is a normal subgroup
negative_passages.docid: 146028
negative_passages.text: $GL(n, \mathbb{C})$ is isomorphic to a subgroup of $GL(2n, \mathbb{R})$
negative_passages.docid: 41237
negative_passages.text: $H$ is normal subgroup of $G$ iff $gHg^{-1}=H$ for every $g$ of $G$
negative_passages.docid: 51352
negative_passages.text: Subgroups of a finite abelian group
negative_passages.docid: 58033
negative_passages.text: $G\times H\cong G$ with $H$ non-trivial
negative_passages.docid: 72251
negative_passages.text: $g\circ f$ is Surjective and $g$ is Injective then Prove that $f$ is surjective
negative_passages.docid: 7575
negative_passages.text: Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$
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: 4838
negative_passages.text: For $N\unlhd G$ , with $C_G(N)\subset N$ we have $G/N$ is abelian
negative_passages.docid: 41231
negative_passages.text: Subgroups of (Z,+) of order n
negative_passages.docid: 29996
negative_passages.text: Prove that if a normal subgroup $H$ of $ G$ has index $n$, then $g^n \in H$ for all $g \in G$
negative_passages.docid: 156941
negative_passages.text: Subgroups of $\mathbb{Z}$