Exploring Coset Multiplication: Is it like Modding Out by N?

[cap.r]

This isn't exactly a homework problem I just want to clarify things for myself.

I am looking back at my algebra notes and I am seeing cosets and factor groups in a new way.

If N is a normal subgroup of G, then the set of left cosets of N forms a group under the coset multiplication given by

aNbN = abN
for all a,b G.

so it looks like we are taking any expression in G let's say a(b+c)+d^2... and taking out whatever is in N and just putting in N. this looks very much like moding out by n in number theory.

is this true? i don't really see a problem with it but I want to make sure there are no counter examples

 

[lippyka]

. Yes, it is true that moding out by N in number theory is similar to taking out whatever is in N and replacing it with N in the context of cosets and factor groups. However, there are some key differences between the two operations. For example, in the number theory context, when you mod out by N, the result is an integer, whereas in the coset multiplication context, the result is not necessarily an element of G. Additionally, although both operations involve a division-like process, in the number theory context, division is exact, whereas in the coset multiplication context, division may not be exact. That is, the coset multiplication may yield an element that is not in G.