Rigid Groups
Suppose that C1,...,Cn are conjugacy classes of a finite group G, and A be the set of n-tuples (g1,...gn) of G such that gi is in Ci and the product g1...gn 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 Ci.)
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 Sn, 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.
Read more about this topic: Inverse Galois Problem
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