But is 2Z + 1 a coset of H?

That is the confusing part to me. I was under the impression that a coset of H had the form aH, not a + H.

Aha! Never fear...almost

*every* person starting group theory has this trouble (myself included).

Really, that's bad notation but a necessary evil.

You'll remember that a group \(\displaystyle (G,*)\) is a non-empty set \(\displaystyle G\) along with an associative binary operation (function) \(\displaystyle *:G\times G\to G\) such that there is some \(\displaystyle e\in G\) with \(\displaystyle *(e,g)=*(g,e)=g\) for all \(\displaystyle g\in G\). For each \(\displaystyle g\in G\) there is some \(\displaystyle h\in G\) such that \(\displaystyle *(g,h)=*(h,g)=e\). And closure (but that is implicit in how we defined \(\displaystyle *\).

That said NO ONE wants to keep writing \(\displaystyle *\) everytime. So, instead of \(\displaystyle *(g,h)\) we write \(\displaystyle gh\). This does

__not__ mean multpilcation. This is just short hand notation for the function \(\displaystyle *\). Remember, \(\displaystyle *\) can be many, many things (function composition, multplication, matrix multiplication, addition, etc.). Now, you'll also remember that in general \(\displaystyle *(g,h)=gh\ne hg=*(h,g)\) and groups for which all the elements satisfy that aer called

*abelian*. Now, it is customary (don't ask me why) to switch from using the short hand notation \(\displaystyle gh\) to \(\displaystyle g+h\) even though it means the same thing when the group is abelian.

So, it is true that a coset is of the form \(\displaystyle gH\) but that really means \(\displaystyle \left\{*(g,h):h\in H\right\}\) and for the group \(\displaystyle \mathbb{Z}\) we have that \(\displaystyle *\) means addition.