Relations between the degrees of transitive constituents of G1 and the absolutely irreducible constituents of permutation representation G* of the group G

Abstract

Let be a finite set of arbitrary elements, G be permutation group on Ω, Δ ⊆ Ω, G1 = {1} = Δ1, Δ2,..., Δn are n orbits of G, ni = |Δi|, fi is the degree of different irreducible representation of D1…Dr appearing in G*, V=V(G) be the ring of all the matrices of G, dimV=k. And also if the irreducible constituents of permutation representation G* are all different, then the expression is a rational integer.