query: Let $G$ be a group and $H$ is subgroup of $G$.
positive_passages.docid: 156463
positive_passages.text: The complement of a proper subgroup generates the whole group
negative_passages.docid: 154878
negative_passages.text: Prove that $H$ is a normal subgroup in the finite group $G=\langle S\rangle $ iff $sHs^{-1}$ $\subseteq H$ for all $s\in S$.
negative_passages.docid: 56245
negative_passages.text: Let H be a subgroup of a group G. Prove that the following statements are equivalent.
negative_passages.docid: 56650
negative_passages.text: Prove that $(G, \circ)$ is a group if $a\circ x = b$ and $x\circ a = b$ have unique solutions
negative_passages.docid: 87882
negative_passages.text: If $N \lhd G$ is abelian and $G/N$ is abelian then $G$ is abelian
negative_passages.docid: 15276
negative_passages.text: $G \times H \cong G \times K$ , then $ K \cong H$
negative_passages.docid: 154398
negative_passages.text: Show that H is a subgroup of G
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: 37971
negative_passages.text: Prove that the center of a group is a normal subgroup
negative_passages.docid: 42891
negative_passages.text: Prove that $Z(G)$ which is the center of $G$ is a subgroup of $G$
negative_passages.docid: 55350
negative_passages.text: H is a subgroup of G and G' is a subgroup of H. Prove H is normal in G.
negative_passages.docid: 61379
negative_passages.text: Let $G$ be abelian, $H$ and $K$ subgroups of orders $n$, $m$. Then G has subgroup of order $\operatorname{lcm}(n,m)$.
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: 39938
negative_passages.text: Subgroups of a cyclic group and their order.
negative_passages.docid: 85011
negative_passages.text: Prove that $\exists a,g,h\in G\colon h=aga^{-1}, g\neq h ,gh=hg$ in a finite non-abelian group $G$.
negative_passages.docid: 63162
negative_passages.text: Is there a characterization of groups with the property $\forall N\unlhd G,\:\exists H\leq G\text{ s.t. }H\cong G/N$?
negative_passages.docid: 35800
negative_passages.text: If $K \leq H \leq G$, show that $[G:K] = [G:H][H:K]$.