In this paper, the solutions of Frobenius equation are investigated in the large Mathieu groups M22 and M23 where n is a positive integer, and are conjugacy classes. Furthermore, three algorithms in GAP system are provided to compute the number of solutions and solutions itself (if exists) for this equation.

Background:       In this work, we compute the decomposition matrix to the spin (projective) characters of , which is the correlations between the irreducible spin characters and the irreducible modular spin characters of , for a given field characteristic of . We may obtain it by figuring out all irreducible spin characters for ,  by fixing all bar partitions, as well as all irreducible modular spin characters for , , where we generate projective character for  by projective character of  and used Maple program to see all the possible of columns to choose the possible the right columns of them. The aim of this study is to pave the way for finding general relationships and theorems to study irreducible modular spin characters. Materials and Methods:    We have  used the -inducing to generate projective character  by projective character of  and  Maple program to choose the possible the right columns Results:    We find decomposition matrix to the spin (projective) characteristics of  for a given field characteristic of  which equals   …   Conclusions:     We have conducted multiple studies to get sufficient data to identify new characteristics and theorems if the field characteristic is prime because there is no standard approach for researching the issue, especially when we prove the field and the change of groups. Prior scholars achieved this when they looked at the division matrix in the field where the characteristic is 0, We  used the -inducing also, we used Maple programming to view every possible columns.

In this work, we compute decomposition matrix for the spin characters , connected between irreducible spin characters and irreducible modular spin characters, when the field characteristic equal to 7. The method used in this work is -inducing in a way to generate projective character for by projective character of and used maple program to see all the possible of columns and then choose the possible the right columns of them. The aim of this research is to pave the way for finding general relationships and theorems to study irreducible modular spin characters.