Galois’s work on permutations Mathematic Algebra
Prominent among Galois’s seminal ideas was the clear realization of how to formulate precise solvability conditions for a polynomial in terms of the properties of its group of permutations. A permutation of a set, say the elements a, b, and c, is any re-ordering of the elements, and it is usually denoted as follows:PermutationThis particular permutation takes a to c, b to a, and c to b. For three elements, as here, there are six different possible permutations. In general, for n elements there are n! permutations to choose from. (Where n! = n(n − 1)(n − 2)⋯2∙1.) Furthermore, two permutations can be combined to produce a third permutation in an operation known as composition. (The set of permutations are closed under the operation of composition.) For example,Composition of permutations
Here a goes first to c (in the first permutation) and then from c to b (in the second permutation), which is equivalent to a going directly to b, as given by the permutation to the right of the equation. Composition is associative—given three permutations P, Q, and R, then (P * Q) * R = P * (Q * R). Also, there exists an identity permutation that leaves the elements unchanged:Permutation identityFinally, for each permutation there exists another permutation, known as its inverse, such that their composition results in the identity permutation. The set of permutations for n elements is known as the symmetric group Sn.
The concept of an abstract group developed somewhat later. It consisted of a set of abstract elements with an operation defined on them such that the conditions given above were satisfied: closure, associativity, an identity element, and an inverse element for each element in the set.
This abstract notion is not fully present in Galois’s work. Like some of his predecessors, Galois focused on the permutation group of the roots of an equation. Through some beautiful and highly original mathematical ideas, Galois showed that a general polynomial equation was solvable by radicals if and only if its associated symmetric group was “soluble.” Galois’s result, it must be stressed, referred to conditions for a solution to exist; it did not provide a way to calculate radical solutions in those cases where they existed.
Prominent among Galois’s seminal ideas was the clear realization of how to formulate precise solvability conditions for a polynomial in terms of the properties of its group of permutations. A permutation of a set, say the elements a, b, and c, is any re-ordering of the elements, and it is usually denoted as follows:PermutationThis particular permutation takes a to c, b to a, and c to b. For three elements, as here, there are six different possible permutations. In general, for n elements there are n! permutations to choose from. (Where n! = n(n − 1)(n − 2)⋯2∙1.) Furthermore, two permutations can be combined to produce a third permutation in an operation known as composition. (The set of permutations are closed under the operation of composition.) For example,Composition of permutations
Here a goes first to c (in the first permutation) and then from c to b (in the second permutation), which is equivalent to a going directly to b, as given by the permutation to the right of the equation. Composition is associative—given three permutations P, Q, and R, then (P * Q) * R = P * (Q * R). Also, there exists an identity permutation that leaves the elements unchanged:Permutation identityFinally, for each permutation there exists another permutation, known as its inverse, such that their composition results in the identity permutation. The set of permutations for n elements is known as the symmetric group Sn.
The concept of an abstract group developed somewhat later. It consisted of a set of abstract elements with an operation defined on them such that the conditions given above were satisfied: closure, associativity, an identity element, and an inverse element for each element in the set.
This abstract notion is not fully present in Galois’s work. Like some of his predecessors, Galois focused on the permutation group of the roots of an equation. Through some beautiful and highly original mathematical ideas, Galois showed that a general polynomial equation was solvable by radicals if and only if its associated symmetric group was “soluble.” Galois’s result, it must be stressed, referred to conditions for a solution to exist; it did not provide a way to calculate radical solutions in those cases where they existed.
