Emrah YAKA

**ABSRACT**
**FINITARY PERMUTATIONS AND LOCALLY FINITE GRAPHS**
Yaka, Emrah
M.Sc. , Department of Mathematics
Supervisor: Assoc. Prof. Dr. Mahmut Kuzucuoglu
December 1998, 104 pages
In this thesis, we studied two different subjects, namely, groups with permuting with finitely many of its conjugates and groups of finitary permutations. In the first part of the thesis, we made a survey of some results of V.E. Kislyakov about groups with an element a permuting with finitely many of its conjugates. In order to understand the structure of such groups, we constructed a graph G_aas follows: The vertex set V(G_a ) = Cl_G(a)and the edge set E( G_a) = {{ x,y}} \| x # y , xy = yx} Here the graph G_a becomes a locally finite graph and the group G becomes a vertex transitive automorphism group of the graph G_a. So by using the properties of locally finite graphs and automorphism groups of locally finite graphs we have the following results: Under the same assumption for G, a² = 1 and *<a,g^-1ag>* is finite for all 4#4, then If G_a has no cycles of length 3, then G_a is finite. If C_G(a) contains finitely many involutions, then G_a is finite. In the second part of the thesis, we made a survey of some results of P.M. Neumann about the transitive finitary permutation groups. Neumann classified these groups into two types, those which are almost primitive and those which are totally imprimitive. By using this classification we can give more details about the structure of these groups. We say G is almost primitive if there existss a maximal congruence of finite modulus, say r_max. And let G is a r_max -class. Then we define the factor group H := G_{G} G_(G)and the cardinal m := \| W /r_max \| , in other words, number of r_max -classes. Then G is called almost primitive group of type ( m , H ). If G is almost primitive group of type ( m , H ), then *W' = < G = < W* where W = HwrSF_m is the main result of Neumann's work. Key word : graph theory, automorphism groups of graphs, finitary permutations groups.