A History of Non-Euclidean Geometry pp 327-381 | Cite as

# Groups of Transformations

Chapter

## Abstract

The group concept was first defined for a certain class of concrete groups, namely

*groups of substitutions,*which were studied in connection with attempts to obtain solutions in radicals of algebraic equations of degree*n*≥ 5. Permutations of roots of algebraic equations were first studied by J. L. Lagrange in his*Reflections on the solution of equations*(Réflexions sur la resolution des éxquations. Berlin, 1771)*[298, vol. 3, pp. 205–515].*Lagrange noticed that if*x*1,*x*_{2},*x*_{3}are the roots of a cubic equation, then each of the cubic radicals in the*Cardano form*can be written as 1/3*(x*_{1}+*ωx*_{2}*+*ω^{2}*x*_{3}), where ω is a cube root of 1. Since the function (*x*_{1}+*ωx*_{2}+*ω*^{ 2 }*x*_{3})^{2}takes on two values under all possible permutations of the roots, it follows that this function is a root of a quadratic equation whose coefficients are rationally expressible in terms of the coefficients of the given equation. Lagrange also noticed that in the case of the fourth-degree equation the function*x*_{1}*x*_{2}+*x*_{3}*x*_{4}of the four roots of this equation takes on only three values as a result of all permutations of the roots and is therefore a root of a cubic equation whose roots are rationally expressible in terms of the coefficients of the given equation. He called this patternthe true principle, and, so to say, the metaphysics of the solution of an equation of third and fourth degree

[298. vol. 3, p. 357].

### Keywords

Manifold Soliton Sine Vasil Actual Element## Preview

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## Copyright information

© Springer Science+Business Media New York 1988