query: Let $H$ be a subgroup of $G$ and let $a$ and $b$ belong to $G$. Then $aH=bH$ or $aH \bigcap bH =\emptyset$
positive_passages.docid: 144788
positive_passages.text: Cosets of a subgroup do not overlap
negative_passages.docid: 27728
negative_passages.text: Question on a homomorphism of a set G.
negative_passages.docid: 93666
negative_passages.text: Does the closedness of a subgroup $H$ of $G$ imply that for $h\in H, g\in G \setminus H : hg \not \in H$?
negative_passages.docid: 35353
negative_passages.text: Isomorphic quotient groups $\frac{G}{H} \cong \frac{G}{K}$ imply $H \cong K$?
negative_passages.docid: 71232
negative_passages.text: if $A$ is Abelian group , $B$ is subgroup of $A$ , Is $B \times A/B \cong A$?
negative_passages.docid: 15276
negative_passages.text: $G \times H \cong G \times K$ , then $ K \cong H$
negative_passages.docid: 141043
negative_passages.text: $H$ normal in $G$. Need $G$ contain a subgroup isomorphic to $G/H$
negative_passages.docid: 98765
negative_passages.text: Showing that $G/(H\cap K)\cong (G/H)\times (G/K)$
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: 130455
negative_passages.text: If $H,K⊲G$ and $H∩K = \{1_G\}$, then all elements in $H$ commute with all elements in $K$
negative_passages.docid: 43759
negative_passages.text: Prove that the set $\mathrm{Aut}(G)$ of all automorphisms of the group $G$ with the operation of taking the composition is a group
negative_passages.docid: 5042
negative_passages.text: Order of a product of subgroups. Prove that $o(HK) = \frac{o(H)o(K)}{o(H \cap K)}$.
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: 39469
negative_passages.text: If $P \leq G$, $Q\leq G$, are $P\cap Q$ and $P\cup Q$ subgroups of $G$?
negative_passages.docid: 148881
negative_passages.text: Aut(GxH) isomorphic to Aut(G)xAut(H)
negative_passages.docid: 81116
negative_passages.text: $gh = hg, \ \gcd(|g|, |h|) = 1\Rightarrow|gh| = |g||h|\ \ (|a|$ = order of element $a)$
negative_passages.docid: 106927
negative_passages.text: How do I prove $[G:H\cap K]\leq [G:H][G:K]$?