query: If |G/H|=4 then G is union of three proper subgroups
positive_passages.docid: 151727
positive_passages.text: If $|G/H|=4$ then $G$ is union of three proper subgroups
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: 151130
negative_passages.text: When is the product of $n$ subgroups a subgroup?
negative_passages.docid: 134353
negative_passages.text: A group of order $108$ has a proper normal subgroup of order $\geq 6$.
negative_passages.docid: 144988
negative_passages.text: Commutator Subgroup of Direct Product equals the Direct Product of the Commutator Subgroups
negative_passages.docid: 102979
negative_passages.text: If $[G:H]$ and $[G:K]$ are relatively prime, then $G=HK$
negative_passages.docid: 63780
negative_passages.text: to show characteristic subgroup is normal in G
negative_passages.docid: 103423
negative_passages.text: Normal subgroup where G has order prime
negative_passages.docid: 77221
negative_passages.text: Prove that $G$ has a subgroup isomorphic to $G/H$.
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: 76943
negative_passages.text: Existence of subgroup of order six in $A_4$
negative_passages.docid: 129452
negative_passages.text: Groups $G,H$ with surjective homomorphisms that are quotient of each other
negative_passages.docid: 156319
negative_passages.text: Show that a group with $|G| = 33$ contains an element of order $3$
negative_passages.docid: 9232
negative_passages.text: Group with proper subgroups infinite cyclic
negative_passages.docid: 103829
negative_passages.text: Find the subgroups of A4
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: 99999
negative_passages.text: A question on identifying normal subgroups of $S_4$