# Talk:Coset Product is Well-Defined

Coset product should be distinguished from subset product by separate notation (at the moment both are denoted with $\circ$) because this result is established by Coset Product of Normal Subgroup is Consistent with Subset Product Definition. I'd propose to denote first with $\circ$ and second by juxtaposition; other propositions are warmly welcome! (maybe $\circ$ and $\bullet$ respectively!?) BTW. In some places cosets are denoted like $aH$ and in some places like $a \circ H$, this is little bit inconsistent… joel ^{talk} 18:15, 29 December 2012 (UTC)

- What's wrong with mentioning that the same notation is used, with reference to the result that they are compatible? --Lord_Farin (talk) 18:59, 29 December 2012 (UTC)

- I think it won't be as clear as with above distinction: in coset space you can introduce many
*different*operations, coset product is one of them. But please notice that subset product (complex product) is defined for any non-empty subgroup subset (which is called a complex) and as such is more general. It just happens that if one restricts himself to complexes which are (left/right) cosets then the result of their complex product is also (the same kind of) coset. Only then those operations coincide. joel^{talk}19:25, 29 December 2012 (UTC) P.S. Coset product is well-defined iff factor subgroup is normal; subgroup product is always well-defined—this also shows that those are different.

- I think it won't be as clear as with above distinction: in coset space you can introduce many

- There is no difference between coset product and subset product but that the former is of an element and the latter is a set. The distinction is mentioned in one of the pages but it's so trivial it's barely even worth discussing. I see no reason to change our approach. --prime mover (talk) 21:37, 29 December 2012 (UTC)

- I'll try once again to convince you! :) To be strict: when speaking about left cosets both products give the same result—left coset (the same for the right cosets); both results can be of course treated as elements of quotient group because its elements are cosets. Difference is that coset product is defined only for elements of coset space (cosets as the name suggest) whereas subset product is defined for elements of power set (subsets as the name again suggest). The question you might ask now is: why bother? (Why bother about subset product for non-coset?, that is.) The reason is that you can describe group products by means of subset products (and normal subgroup) which is impossible with coset product alone which is useful only for quotient group. I hope you feel more convinced by now! :D joel
^{talk}21:12, 30 December 2012 (UTC)

- I'll try once again to convince you! :) To be strict: when speaking about left cosets both products give the same result—left coset (the same for the right cosets); both results can be of course treated as elements of quotient group because its elements are cosets. Difference is that coset product is defined only for elements of coset space (cosets as the name suggest) whereas subset product is defined for elements of power set (subsets as the name again suggest). The question you might ask now is: why bother? (Why bother about subset product for non-coset?, that is.) The reason is that you can describe group products by means of subset products (and normal subgroup) which is impossible with coset product alone which is useful only for quotient group. I hope you feel more convinced by now! :D joel

- Nope, sorry. --prime mover (talk) 21:20, 30 December 2012 (UTC)

- Same here. It's still the case that the one is a restriction of the other; and while they are conceptually different, I think it only confusing to denote them by different symbols. Many people would be unhappy to forcefully refer to e.g. a crafted evil like $\left({f \restriction_T}\right) \restriction_V$ only to invoke a theorem stating that it equals $f \restriction_V$ (in case $V \subseteq T \subseteq \operatorname{dom} f$). It obfuscates what is to be conveyed. --Lord_Farin (talk) 22:21, 30 December 2012 (UTC)

- I think I see your point. joel
^{talk}

- I think I see your point. joel

Another question: should be proved here that coset product is well defined iff subgroup is normal? joel ^{talk} 00:22, 31 December 2012 (UTC)

- On another page, if it has not already been done. --prime mover (talk) 06:33, 31 December 2012 (UTC)

- Reading definition of the coset product I see you defined it for cosets modulo normal subgroup only… The theorem I mentioned has sense if coset product is defined for every coset (as in its definition); then it'd be advisable to put this result as a part of mentioned theorem. What do you think? joel
^{talk}15:43, 31 December 2012 (UTC)

- Reading definition of the coset product I see you defined it for cosets modulo normal subgroup only… The theorem I mentioned has sense if coset product is defined for every coset (as in its definition); then it'd be advisable to put this result as a part of mentioned theorem. What do you think? joel

## H subgroup of G. Left coset multiplication is well-defined by (aH)(bH) = (ab)H iff H is normal subgroup of G.

Can't we make these proofs iffs because they actually are?

- no --prime mover (talk) 15:46, 3 February 2014 (UTC)

- While what you suggest is indeed a way which we could go, the main thrust of this page is merely to indicate that coset product is well-defined. Restructuring this page is a lot of work which I don't want to do.

- The proof you suggest may be added as a separate page. --prime mover (talk) 17:26, 3 February 2014 (UTC)