Some Properties Of Two Classes Of Finite Groups

The groups that consist of non-bijective transformations on a non-empty set A where the binary operation on a group G is the composition of transformations have been recently given. A permutation group on a set A is a group which consisting of bijections from A to A with respect to compositions of mappings have been made in the framework of the abstract algebra approach. In this thesis, we show that there exist some groups which consist of non-bijective transformations on a non-empty set A. We introduce the maximal order of these groups (that consisting of transformations on a non-empty set A and the group has no bijection as its element). We found the order of these groups is not greater than (n-1)! and any kind of these group isomorphic to a permutation group on a quotient set of A with respect to an equivalence relation on A. Moreover, we answer the question that if G=UV with U, V are subnormal in G then it holds that the maximal normal χ-subgroup of G is equal UχVχ. We found if p is a prime and take the SHP-class X to be the class of all finite p-group, then for any finite group Gχ will be OP(G). We prove that U, V are both subnormal in G and Op(G)=〈y〉and OP(U)=OP(V)= 1. In particular, OP(G)≠Op(U)Op(V) if p and q are two primes such that q≡1(mod/p) with N is a cyclic group of order q, H is an elementary abelian group of order p2, and〈x〉 act on N faithfully and〈y〉act on N trivially on the semi-direct product of N and Hwhere U= N(x)and V=〈xy〉. In addition, we consider the groups that consist of non-bijective transformations on a non-empty set A by using the fixed points. We found the order of these groups is equal of the order of orbit multiply order of stabilizer for any fixed points.We divide the thesis to four chapters. First chapter, study a new group that consists of non-bijective transformations on a non-empty set A where the binary operation on group G. These new groups are given by examples. And, we consider the maximal order of a group consisting of transformations on a non-empty set A and the group has no bijections as it is elements. The second chapter, we review some basic concepts in a finite group such a binary operation on a group G and some other definitions. Moreover, we study the symmetric groups and the proof of theorems and lemmas that have used in this thesis. In the third chapter, we consider new groups which consists of non-bijective transformations on a non-empty set A and introduce the maximal order of groups which consist of non-bijective transformations on a non-empty set A of order n where the binary operation on a group G as a new result. The maximal order of these groups is (n-1)!. We apply these groups to study some properties of two classes of finite groups. Fourth chapter, devoted to study our groups by using the fixed points.