Linearly Equivalent Actions of Solvable Groups

Let G be a group. A G-set is a finite set provided with a left action of g 4 G. For a G-set X and g g G we write X s x g X : gx s x . The permutation character of a G-set X is the function p from G to the ring X Z of integers that sends g to the cardinality aX g of X . We say that two G-sets X and Y are linearly equï alent if p s p . By representation X Y theory of finite groups, two G-sets X and Y are linearly equivalent if and only if the permutation modules C X and C are isomorphic as modules w x over the group ring C G ; here C denotes the field of complex numbers.