query: If $G = H \cup K \cup L$, then $[G:H] = [G:K] = [G:L]=2$
positive_passages.docid: 152589
positive_passages.text: How to show that if $G$ is union of three proper subgroups, then there is an epimorphism from $G$ to the Klein 4-group?
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: 45760
negative_passages.text: if G is generated by {a,b} and ab=ba, then prove G is Abelian
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: 111763
negative_passages.text: True or False $A - C = B - C $ if and only if $A \cup C = B \cup C$
negative_passages.docid: 141683
negative_passages.text: $ |G_1 |$ and $|G_2 | $ are coprime. Show that $K = H_1 \times H_2$
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: 157840
negative_passages.text: Is it true that $(R\times S)[G]\cong R[G]\times S[G]$?
negative_passages.docid: 151047
negative_passages.text: a non trivial example: $G/H$ is isomorphic to $G$
negative_passages.docid: 144732
negative_passages.text: If $[G:H]=n$, then $g^{n!}\in H$ for all $g\in G$.
negative_passages.docid: 77361
negative_passages.text: How to prove $|H|[K:K \cap (g^{-1}Hg)]=|K|[H:H \cap (gKg^{-1})]$?
negative_passages.docid: 122581
negative_passages.text: Prove or disprove that $G_1/H_1 \cong G_2/H_2$
negative_passages.docid: 49768
negative_passages.text: If Gal(K,Q) is abelian then |Gal(K,Q)|=n
negative_passages.docid: 70307
negative_passages.text: $(A\cap B)\cup C = A \cap (B\cup C)$ if and only if $C \subset A$
negative_passages.docid: 76823
negative_passages.text: Is it the case that for all sets $A, B, C,$ and $D$, $(A \times B) \cup (C \times D) = (A \cup B) \times (C \cup D)$?
negative_passages.docid: 13253
negative_passages.text: Prove $f(S \cup T) = f(S) \cup f(T)$
negative_passages.docid: 105951
negative_passages.text: Does $G\times K\cong H\times K$ imply $G\cong H$?