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I'm working--slowly--through Herstein's Topics in Algebra, and I've hit on a lemma that I'm having trouble visualizing: There is a 1-1 correspondence between any two right cosets of H in G (2.7, p. 35 my edition).

It's easy for me to make sense of this in the case of an infinite G, as it's describing modular arithmetic.

However, I'm having trouble visualizing an example where this would hold with finite H and G.

It would seem possible to create a subgroup H that was, in other ways, a valid subgroup, but which did not have a number of members divisible by the number of members in the group G (e.g. a subgroup with three members in a group with ten)

Can someone help me make more sense of what's going on here?

Rich Jensen
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    It's saying, in effect, that every coset of $H$ has the same number of elements as $H$. – Angina Seng Apr 26 '18 at 16:10
  • This is nothing like modular arithmetic in general for infinite groups. – Tobias Kildetoft Apr 26 '18 at 16:19
  • @TobiasKildetoft Could you clarify what you mean? My understanding is that if you have an infinite subgroup H, e.g. {...,-12, -8, -4, 0, 4, 8, 12,...} Your cosets in G will be {...,-13, -9, -5, -1, 3, 7, 11, ...}, {...-14, -10, -6, -2, 2, 6, 10,...}, etc. And in this case there is a 1:1 relationship between each coset and H, and that the members of each coset are an equivalence class. The intersection of H and its cosets is a null set, and the union of H and its cosets is G. This seems to describe the underlying structure of modular arithmetic. – Rich Jensen Apr 26 '18 at 18:38
  • But the integers is not the only infinite group. Sure, if $G = \mathbb{Z}$ then you get modular arithmetic. – Tobias Kildetoft Apr 26 '18 at 18:40
  • Actually, this is extremely misleading, since in this case, every collection of cosets is a group, since all subgroups are normal in an abelian group. – Andres Mejia Apr 26 '18 at 18:46
  • @TobiasKildetoft Thanks for the clarification. That was indeed where I was coming from. Herstein makes the relationship explicit a few paragraphs earlier--Sorry I wasn't more exact in my original question. I'll have to blame it on my degree in the humanities. :D – Rich Jensen Apr 26 '18 at 18:48
  • @AndresMejia I don't follow that. What do you mean by a collection of cosets being a group in this case? – Tobias Kildetoft Apr 26 '18 at 18:48
  • I mean that in general, one considers left cosets, but these cases are "modular arithmetic" because when the group is normal, the collection of left/right cosets inherit a natural group structure. I just think this image makes cosets seem much more like "modular arithmetic" than they should. I guess I'm being unclear, so I'll stop speaking now :). – Andres Mejia Apr 26 '18 at 18:49
  • @AndresMejia, it seems like a good entry point if you're someone like me, and you have a hard time imagining specific examples. :D – Rich Jensen Apr 26 '18 at 19:00

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