Classifacation of Sylow p-subgroups of general linear groups

This thesis, we studied on finding p-sylow subsgroups of some spesific groups psubsgroupsare very important in the classification of finite groups.İn 1872, Peter Ludind Sylow proved three important theorems in group theory.After that these theorems called a Sylow Theorems. These theorems gives thatsome subgroups in finite groups and give information. Lagrange Theorem saysthat every order of finite subgroup divides the order of groups. That inverse is nottrue. İn other words, sylow theorems gives that given k integer and p prime ifdivides the order of G then G has a subgroup of order . İn this with studyon finding p-sylow subgroup of Sym(k) and In a finite group G, it iscalled sylow p-subgroup of G. Given any finite set which has n elements. Allpetmutations of this set is called Sym(n). Let and p a prime.Then a Sylow p-subgroup of is isomorphic towhere P is a Sylow p-subgroup of and P acts on bypermuting the components. As usual by we mean the finite field of order q,the group of square matrices over of size n nonzero determinant. ASylow p-subgroup of is of the form forand where k is the multiplicative order of q modulop and i s the cyclic group of order .Key words: Sylow p-subgroups, symmetric groups, general linear groups, semidirect product, wreath product