Lie groups were initially introduced as a tool to solve or simplify ordinary and partial differential equations. The model for this application was Galois’ use of finite groups to solve algebraic equations of degree two, three, and four, and to show that the general polynomial equation of degree greater than four could not be solved by radicals. In this chapter we show how the structure of the finite group that leaves a quadratic, cubic, or quartic equation invariant can be used to develop an algorithm to solve that equation.

Marius Sophus Lie (1842–1899) embarked on a program that is still not complete, even after a century of active work. This program attempts to use the power of the tool called group ...

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