https://doi.org/10.31871/WJRR.15.5.15 World Journal of Research and Review (WJRR) ISSN:2455-3956, Volume-15, Issue-5, November 2022 Pages 01-03 1 www.wjrr.org Abstract- This paper is set to compute the ranks and the subdegrees of acting on the cosets of its subgroups namely; and . This is done using the method proposed by Ivanov et al. which uses marks of a permutation group. For the action of on the cosets of the subdegrees are shown to be and and the rank is . For the subdegrees are and and the rank is . Index Terms- Rank,Subdegrees I. INTRODUCTION a) Burnside’s definition of marks Let be a permutation representation (transitive or intransitive) of a group on X. The mark of the subgroup H of G in is the number of points of X fixed by every permutation of H. incase (/ ) i G G is a coset representation ; , the mark of in (/ ) i G G is the number of cosets of in left fixed by every permutation of . b) White’s definition of marks; where (statement) a) Ivanov et al. definition of marks Ivanov et al. (1983) defined the mark in terms of normalizers of subgroups of a group as; if and is a complete set of conjugacy class representatives of subgroups of that are conjugate to in G, then; Stanley Rotich1*,Fidelius Magero2 Kangogo Moses2, Ireri Kamuti2 1. Department of Mathematics Statistics and Actuarial Science , Machakos University, P. O. Box 136-90100, Machakos, Kenya 2. Mathematics Department, Kenyatta University, P. O. Box 43844-00100, Nairobi, Kenya In particular when is conjugate in to all subgroups that are contained in and are conjugate to in G and; It can be shown that these definitions are equivalent. (Kamuti,1992) A. Lemma Let (d coprime to p) be a cyclic subgroup of order , then; sign as . (Dickson, 1901) B. Lemma Let be a cyclic subgroup of order p in G, then (Dickson, 1901) C. Lemma Let be a divisor of and be the quotient, then (Dickson, 1901) D. Theorem The number satisfy the system of equations; for each . (Kamuti, 1992) II. SUBDEGREES OF ON THE COSETS OF Since H is abelian, each of its subgroups is normal. Suppose H has s subgroups say, , with and, RANKS AND SUBDEGREES OF ON THE COSETS OF ITS SUBGROUPS Stanley Rotich 1* ,Fidelius Magero 2 Kangogo Moses 2 , Ireri Kamuti 2