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Good Math by Mark C. Chu-Carroll

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The Roots of Symmetry

Galois and Abel independently discovered the basic idea of symmetry. They were both coming at the problem from the algebra of polynomials, but what they each realized was that underlying the solution of polynomials was a fundamental problem of symmetry. The way that they understood symmetry was in terms of permutation groups.

A permutation group is the most fundamental structure of symmetry. As we’ll see, permutation groups are the master groups of symmetry: every kind of symmetry is encoded in the structure of the permutation group.

Formally, a permutation group is a simple idea: it’s a structure describing all of the possible permutations, or all of the possible ways of rearranging the elements of a set. Given a set ...

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