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matt grime
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radou said:hence [itex]H \cap N[/itex] must be a subset of [itex]H \cap N[/itex].
I presume you didn't mean to write that.
radou said:hence [itex]H \cap N[/itex] must be a subset of [itex]H \cap N[/itex].
matt grime said:I presume you didn't mean to write that.
matt grime said:Since, at no point, did you invoke the fact that HN=KN, you might want to rethink what that means.
Why did you ignore my reply? I'll say it again: What are |HK| and |HN|?radou said:If HN = KN, we know [itex]HN\subseteq KN[/itex], which is quite obvious, since H < K, and we know [itex]KN\subseteq HN[/itex], so x = kn e KN implies x e HN, so every k must be from H, and hence [itex]K\subseteq H[/itex], which implies K = H. But why are two facts given in the problem (HK = KN and the first one about intersections), since they lead to the same conclusion? Again, obviously my reasoning is wrong.
morphism said:And your reasoning failed because kn being in HN doesn't imply k must be in H. kn in HN means there exists an h in H and an n' in N such that kn = hn', so k = hn'n-1, which doesn't necessarily lie in H.
morphism said:Why did you ignore my reply? I'll say it again: What are |HK| and |HN|?
morphism said:How is H[itex]\cap[/itex]N related to H?
morphism said:Right. h-1k is in a subgroup of H (and therefore in H). So...
morphism said:Yup.
(message too short)
matt grime said:Any map is surjective onto its image.