# Soft Question: What to do with proofs of nontrivial theorems?

A major part of reading any text in mathematics is understanding theorems and proofs. For me, I would try to find a proof by myself before reading the one given in the text. But sometimes the proof may be too hard for me, e.g., it may involve tricks that I can never think of. So my question is:

What is the best thing to do with these "hard" proofs? When I cannot prove it myself, should I give up and read the proof given in the text, or should I leave it and try to prove it later? Or maybe there are better things to do?

To make this question more concrete (and not too soft), let me say something about my background. I only know some basic mathematical analysis and linear algebra on the freshman level, and theorems there are usually not so hard to prove. Lately I have been reading group theory, working through Rotman's An Introduction to the Theory of Groups. While this book is not difficult to understand, I found quite a few theorems that I could not prove. An example is Theorem 4.8, which states that the number of subgroups of order $p^s$ in a finite $p$-group $G$ (where $p^s\leq|G|$) is congruent to $1$ mod $p$. Another example is P. Hall's theorem on Hall subgroups. The proof is more than one page long and I have no clue about how to tackle it.

I know that there will be more and more nontrivial theorems like these as I study mathematics, therefore I am here to ask what I should do. Since I have been teaching myself all along, I don't want to get it wrong in the first place...

If this question is still too soft or vague, please let me know. And thanks for all advice!

First, kudos on first trying to find a proof on your own. This is an excellent habit and will pay dividends.

However, I don't think you should skip the proof, planning to return later. Some proofs involve new techniques, ingenious twists, even what you might call "strokes of genius". Today's math is the end result of over 2000 years of work by some the smartest people on the planet, and hundreds of thousands of others who weren't slouches either. It's unrealistic to think that you'll be able to match all that.

My personal approach: if I can't see how to prove something, after giving it a decent shot, I skim the proof in the book. Perhaps the ingenious twist will pop out. Then I'll try to finish the proof on my own. But I'll read the proof line by line if I have to. A day or two later, I'll try to reconstruct the proof in my head (preferably while taking a long walk), to see if I've really digested its pivotal ideas.

The downside to saving the proof till later: it can easily snowball, where you miss a key point on page 10, say, and so can't do (or even follow) the proof on page 12, and so on.

Ideally, the exercises in the book will give you plenty of chances to solidify your understanding.

• Thanks! I guess I‘ve been treating the theorems as if they are series of exercises to be solved. :) – Colescu Nov 14 '17 at 4:35
• Yes. Of course, learning math is an active process, which is why first trying for a proof on your own is such a good thing. Side effect: when I do give up and read the proof, I think I understand it better than if I hadn't tried. "Oh, that's how they get around the problem that the space isn't necessarily compact", or whatever. – Michael Weiss Nov 14 '17 at 15:21
• Also, there's the question of whether one should do all the exercises. Pro: obviously a better understanding. Con: takes longer, which means you won't get to the next book on your reading list till later. Life is finite, unfortunately! – Michael Weiss Nov 14 '17 at 15:25
• Couldn't agree more! – Colescu Nov 15 '17 at 4:26

I think one often needs to work out more examples of the Theorem itself, before proceeding to all details in the proof. For example, choose a very explicit example, say $p=2$, $s=2$ and $G=A_4$. Then $G$ has $12$ elements. Determine all $p$-subgroups of order $p^s=2^2=4$, and verify everything explicitly.

• That's a good suggestion, thanks! But does it really hint upon the proof? I think examples may be too specific... – Colescu Nov 11 '17 at 10:30
• You can carry out every step of the proof for a given example. This helps to make every step and every definition explicit. In this sense examples are not too specific. – Dietrich Burde Nov 12 '17 at 8:53