I'm trying to find the right level of explicitness to use when writing algebra proofs. The challenge is that algebra proofs, once complete, can often devolve into simply a series of equations. But adding too much explanation can be distracting.
For example, when working on:
If $N$ is a normal subgroup of $G$ such that $G/N$ is abelian, prove that the commutator subgroup $[G, G]$ is a subset of $N$.
I wrote the proof below, trying to strike the right balance of being clear and explicit, while still being succinct. I tried to justify each step in a way which still kept the compactness of formulation inherent to algebra.
Have I done that well? How could the exposition be improved? (Of course, I should first ask if it is correct.)
Note: Proofs of this fact may be available; this question is about the exposition (and verification) of this proof.
Note 2: Also germane to this question is: How to properly write proofs of this nature - balancing between strings of symbolic equations and words. I considered asking this as a separate question, but concluded that it needed to be discussed in the context of a specific problem. Hence, this question.
Proof: Recall that $x$ is a member of subgroup $N$ if and only if $xN = N$. It therefore suffices to show that $ghg^{-1}h^{-1}N = N$ for all $g, h \in N$. We have:
$$\begin{align*} ghg^{-1}h^{-1}N &= ghg^{-1}h^{-1}NN \quad \text{($N$ is a subgroup)} \\ &= ghNg^{-1}Nh^{-1} \quad \text{($N$ is normal)} \\ &= ghNh^{-1}Ng^{-1} \quad \text{($G/N$ is abelian)} \\ &= ghh^{-1}g^{-1}NN \quad \text{($$N is normal)} \\ & = N. \end{align*}$$
Update
I gather from the initial comments and answer that the answer depends on the anticipated audience. That makese sense.
What would be most helpful is to teach by example: Present an alternate exposition of the proof above, perhaps shorter, perhaps longer, but that illustrates how you'd write this (at least for whatever audience you choose).
There's a meta answer to this question. The right level of explicitness in a proof depends on the reason you are writing the proof.
If it's an exercise in a course, its purpose is to show the instructor that you understand precisely why the theorem is true. Your proof in the question does that. I think strings of symbols are rarely the best way to write proofs, but in this case that works just fine.
If it's a proof in a textbook, or a more interesting course exercise, I think more motivation and more words would be better.
If it's part of a research paper in a journal, "left to the reader" would be appropriate.