Group theory is interesting in part because the constraints on $~$\\bullet$~$ are at a "sweet spot" between "too lax" and "too restrictive\." Group structure crops up in many areas of physics and mathematics, but the group axioms are still restrictive enough to make groups fairly easy to work with and reason about\. For example, if the order of $~$G$~$ is prime then there is only one possible group that $~$G$~$ can be \(up to isomorphism\)\. There are only 2, 2, 5, 2, and 2 groups of order 4, 6, 8, 9, and 10 \(respectively\)\. There are only 16 groups of order 100\. If a group structure can be found in an object, this makes the behavior of the object fairly easy to analyze \(especially if the order of the group is small\)\. Group structure is relatively common in math and physics; for example, the solutions to a polynomial equation form a group under permutation \(a fact from which the unsolvability of quintic polynomials by radicals was proven\)\. Group theory is thus a useful tool for figuring out how various mathematical and physical objects behave\. For more on this idea, see the page on group actions\.

I got lost here -- I feel like I sort of know what "under permutation" means, but can't picture what it means in the context of solutions to polynomials. What exactly is being permuted?

## Comments

Qiaochu Yuan

This statement is just wrong. I will fix it. (The correct statement is that there's a group acting on the solutions called the Galois group; it's the solutions that are being permuted.)