# Concrete examples of abstract algebra delivering simple solutions to hard problems

Often people allude to examples of hard problems or theorems of which trivial solutions (or proofs) were found by applying techniques from abstract algebra. Can you give an example of such a problem or a theorem?

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Solving "rubic's cube"? – amWhy Nov 4 '12 at 15:02
It seems to me that there ought to be a lot more answers to this question. – Mike Jones Apr 3 at 22:54

The product of all the non-zero residue classes modulo a prime $p$ is -1 (mod $p$). This can be proved by noting that $\mathbb Z/p \mathbb Z$ is a field and in multiplying every non-zero number, you can pair off everything in inverse pairs except for -1 and 1.

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Here are some nice applications of group theory. I do not know if solving these problems were hard before the advent of group theory. I know example 4) was possible only due to the development of both group theory and topology.

1) Fermat's Little Theorem: For all $a \in \mathbb{Z}$ and a prime number $p$, $a^p \equiv a (mod \hspace{1mm} p)$.

2) Abel-Ruffini-Galois Theorem (unsolvability of quintic and higher degree polynomials). I added Galois's name because he came up with groups in the first place to solve this problem. I think the result is usually known as the Abel-Ruffini Theorem.

3) How many rotationally distinct colorings are there of a cube with three colors? (Application of Burnside's Lemma)

4) The torus and the $2$-sphere are not topologically equivalent. (Fundamental groups of topological spaces)

5) Which real numbers can be constructed using ruler and compass alone?

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If you can show that two groups are isomorphic, and you can show that from one group has certain properties, then you know that the other (more difficult) group has those properties as well.

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That's not very concrete :D – Alexei Averchenko Nov 4 '12 at 15:07
True. I dont know enough about the subject to give a concrete example right know. I will be thinking about it :) – Applied mathematician Nov 4 '12 at 15:07