Inverse Galois Problem - Rigid Groups

Rigid Groups

Suppose that C₁,...,C_n are conjugacy classes of a finite group G, and A be the set of n-tuples (g₁,...g_n) of G such that g_i is in C_i and the product g₁...g_n is trivial. Then A is called rigid if it is nonempty, G acts transitively on it by conjugation, and each element of A generates G.

Thompson (1984) showed that if a finite group G has a rigid set then it can often be realized as a Galois group over a cyclotomic extension of the rationals. (More precisely, over the cyclotomic extension of the rationals generated by the values of the irreducible characters of G on the conjugacy classes C_i.)

This can be used to show that many finite simple groups, including the monster group, are Galois groups of extensions of the rationals. The monster group is generated by a triad of elements of orders 2, 3, and 29. All such triads are conjugate.

The prototype for rigidity is the symmetric group S_n, which is generated by an n-cycle and a transposition whose product is an (n-1)-cycle. The construction in the preceding section used these generators to establish a polynomial's Galois group.