Journal of Natural Sciences Research www.iiste.org ISSN 2224-3186 (Paper) ISSN 2225-0921 (Online) Vol.8, No.6, 2018 39 On Sylow Subgroups of Permutation Groups M. Bello 1 A. B. Umar 2 Mustapha Danjuma 3 Bulus Simon 3 1.Mathematics and Computer Science Department, Federal University Kashere, Gombe State, Nigeria 2.Department of Mathematical Science, Abubakar Tafawa Balewa University Bauchi, Bauchi State, Nigeria 3.Department of Mathematics and Statistics, Abubakar Tatari Ali Polytechnic, Bauchi State, Nigeria Abstract A research on Sylow Subgroups of permutation groups is carried out in this paper. The research investigates the nature of Sylow subgroups of permutation groups and examines its behaviors. Introduction The concept of Sylow subgroups originate from Sylow theorems which are collection of theorems that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems assert a partial converse to Langrange’s theorem. The research construct different types of permutation groups, which includes symmetric groups, dihedral groups and groups generated by wreath products of two permutation groups and investigates their behavior in terms of normality in the parent group, primitivity and transitivity of the sylow subgroups of these groups. The research adopt the concept of M. Bello et all (2017), work on a numerical search for polycyclic and locally nilpotent permutation groups. Definition 1.1 Let G be a group, and let p be a prime number  A group of order   for some 1 is called a p-group.  A subgroup of   for some 1 is called a p-subgroup.  If ||     where p does not divide m, then a subgroup of order   is called a Sylow p- subgroup of G. The illustrations of the above are given below: (i)   1, 34, 12, 1234, 1324, 1324, 1423, 1423 G is a p-group since, ||  2  . Some ofthe subgroups of G are as follows:    1    1, 12    1, 34    1, 1234    1, 1324    1, 1423    1, 12, 34, 1234 The subgroups   ,  ,  ,  ,  are p-subgroups, since they have order p and   whose order is p-power. (ii)   1, 465, 456, 132, 132465, 132456, 123, 123465, 123456, 142536, 143625, 142536, 163524, 162435, 162435, 152634, 152634, 153426 ||  2  3  . The subgroups of G are as follows:    1    1, 142536    1, 152634    1, 162435    1, 465, 456    1, 123, 132    1, 132465, 123456    1, 123465, 132456  !  1, 142536, 132465, 162435, 123456, 153426  "  1, 152634, 132465, 142536, 123456, 163524    1, 162435, 132465, 152634, 123456, 143625    1, 132456, 123465, 142536, 162435, 152634    1, 465, 456, 123456, 123, 123465, 132465, 132456, 132    1, 465, 456, 132, 132465, 132456, 123, 123465, 123456,