Value of filling the gaps in a proof

Question

I am studying a paper "On Finite Groups with Given Conjugate Types I" recently. The author uses many words like "obviously", "clearly", "trivially", etc. in his proof. But these "obvious" implications are not so obvious for me. I always need to spend a lot of time to prove the "obvious" (if I can prove it). http://bfhaha.blogspot.tw/2014/06/a-note-on-paper-of-ito-on-finite-groups.html

Question: Why is there no man who writes a note of the paper to explain the "obvious"?

My guesses are:

1. Mathematicians think that proving a new result is more important than giving details of the proof.
2. Proving the "obvious" is a good opportunity for readers to practice.

Answer 1

Do you recommend that someone get a shave or do you describe how they should cut each hair?

Obviousness is context-dependent, in particular on the expected/intended audience. I don't need to tell someone how to shave, because to anyone I would tell to do so, it's obvious.

Answer 2

It is often a good exercise to fill in these gaps in proofs.

First, it will greatly increase your understanding of the subject matter.

Second, others may have the same problem, and you might get a publication showing your work.

Finally, there might be an error hidden by the "obviously" and "clearly". Revealing this error cen be useful in a number of ways: you might show that the result is false by constructing a counterexample, or you might be able to fix the error, either by correcting it or by adding some additional conditions that make the error correct.

In any case, go for it- fill in the gaps!