I want to learn the correct notation and language with which to communicate clearly about a quotient on the multiplicative group $G=(\Bbb Q^+,\times)$
In particular, I want to know the notation with which to clearly identify the cosets and the unique indexing elements.
The equivalence relation is $x\sim y\iff\exists i,j\in\Bbb Z:2^i \times3^j\times x=y$
Let $N$ be the 3-smooth numbers then $1\sim n\iff n\in N$ so I think this is a normal subgroup of $G$ and the equivalence class of the identity element.
Then the multiplicative group generated by the primes greater than $3$ is its dual and the quotient group. So I think I would write $G/N=Q$
Then every element of $G$ has a unique representation as $q\times n:q\in Q, n\in N$
All good so far, what I'm not clear on now is how to express (in notation) the unique representatives and their cosets.
Tentatively I think I can write $q\in Q$ and $[q]=q\times N$.
Is $Q\subset G$ or are $[q]\in Q$? i.e. does $Q$ contain singletons or cosets?
Assuming $Q$ contains cosets, how do I identify a (the) set of unique indexing elements, and its elements?
I have in mind the unique set satisfying $\lvert x\rvert_2=1=\lvert x\rvert_3$
Best Answer
First, it is never true formally that $\;Q=G/N\subset G\;$ . The elements of each set are just different.
Next, if you already chose $\;q\in Q=G/N\;$ , then there is no need to use $\;[q]\;$ , which is a notation usually reserved, in this context, to denote equivalence classes for an element in the original set. Thus, if you take $\;g\in G\;$ , then it'd make sense to talk about $\;[g]:=gN\in G/N=Q\;$ (no need of multiplication sign $\;g\times N\;$ , which could be confusing) .
The elements of $\;Q=G/N\;$ are cosets, which is the name given in this context to the equivalence classes in $\;G\;$ we get fro the equivalence relation determined by $\;N\lhd G\;$ .
Thus, a singleton in $\;Q\;$ could be denoted by $\;\left\{\;[g]\;\right\}=\left\{\;gN\;\right\}\;$ , for any $\;g\in G\;$ .
I think you're having some trouble understanding all this, and you chose a rather problematic example of subgroup in a rather problematic group (for a beginner). There are way simpler examples of groups and normal subgroups to work your way through these definitions.