Before group theory was invented, the notion of symmetry was first captured mathematically in what we now call the permutation groups. The study of permutation groups was the reason group theory was originally created!

Group theory was developed as a part of the algebraic study of equations. During the nineteenth century, mathematicians were obsessed with finding ways to compute the roots of polynomials. This obsession reached the point where it became a spectator sport: people would assemble in auditoriums to watch master algebraists race to be the first to compute the solutions to a polynomial! The holy grail to these competitive equation-solvers was a simple equation that could yield the solutions to a polynomial. ...

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