ABSTRACT
The notion of a group originated from ideas about transformations of geometrical objects. The study of groups of symmetry paved the way for the denition of an abstract group. By the end of 17th century, methods of solving quadratic, cubic and biquadratic equations were known. In the attempts to solve a quintic equation, permutation groups proved to be relevant. Lagrange got the idea that groups had something to do with equations. The permutation groups S2, S3 and S4 were ‘well-behaved’ groups (in the sense of solvability) and were associated with a quadratic, cubic and bi-quadratic equation respectively. Lagrange knew that S5 behaved ‘differently’. It was Abel who showed that an equation of the fth degree was not solvable by ‘radicals’. During this period, Everiste Galois discovered a necessary and sufcient condition for an nth degree equation to be solvable by radicals. Galois showed that to each algebraic equation
f (x) = a0xn + a1xn + · · ·+ an = 0 (a0 6= 0, a0,a1, . . . ,an ∈Q) one could attach a group of permutations to the polynomial f (x) of the equation. The equation is solvable by radicals if, and only if, the group associated with the polynomial is solvable. After Galois, Felix Klein (1849–1929) attempted to describe all geometries by their groups of symmetries. This, he called the Erlangen Programme. Since then, group theory has become a major tool in many branches of mathematics.
