Ndirangu, Hannah Wagio2020-01-212020-01-212019-03http://ir-library.ku.ac.ke/handle/123456789/20055A Research Project Submitted to the School of Pure and Applied Sciences of Kenyatta University for the Award of Degree of Master of Science (Pure Mathematics), March 2019Subgroups and supergroups of various symmetric groups have been researched on extensively. Various suggestions by researchers have been provided on how to find the numbers of Sylow -subgroups. Casadio (1990) has provided the proof for the third Sylow theorem, which will greatly contribute to the finding of the possible variety of numbers of the Sylow -subgroups. He stated that the number of the Sylow -subgroups of a group is congruent to 1 modulo and divides i.e. and is gotten by dividing the order of the group with the order of the Sylow -subgroup. The research will be made up of five chapters. It has emphasized the number of subgroups, supergroups, lattice structures and the ascending chains of various symmetric groups. We shall largely study the Sylow - subgroups in symmetric groups , n =1,2,3,4,5,6,7,8,9,10 which will lead to a generalization of the number of Sylow -subgroups in any symmetric group and hence coming up with a formula for getting the number of the Sylow -subgroups of symmetric groups of any given order. Gow (1994) showed that for any symmetric group Sn , where is a prime, the number of Sylow -subgroups is ( p - 2)!. Hence we target on finding the number of the Sylow p -subgroup for any symmetric group, which will be given by;   ( 1)! ( 2)! 0       m p n Where is the number of Sylow p-subgroups of any symmetric group.enThesis