More Notation

I'm going to interrupt our regularly-scheduled discussion of factor groups to inroduce a few notational conventions I've been avoiding that I now think are going to be necessary. First, I've been using Z(n) to denote the cyclic group of order n, but the standard notation is Z_n so that's what I'll be using from now on. (The answer to your next question is that I have no idea why I didn't just introduce it as Z_n to begin with.) Also, in multiplicative groups, I'm going to start leaving out the ∙ most of the time. That is, instead of a∙b I will usually just write ab from now on. Along those same lines, I'm going to start leaving out a lot of parentheses - especially in cosets. Where I would have written (aN)∙(bN) = (a∙b)N I will generally write aNbN = abN in the future. These conventions are not just my laziness - I do it that way because that's the way everyone else does it. I will admit, though, that the reason that these things have become convention - and the reason behind most mathematical convention - is because mathematicians before me are lazy.