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6 ( alois theory) ( ) Joseph Louis Lagrange ( ) Augustine Louis Cauchy Ludwid Sylow ( ) Camille Jordan Milter Shur Burnside Helpert Frabinous 6

7 (Class of finite super soluble groups) (Class of abelian finite groups) (Class of finite generated nilpotent groups) (Composition series) (Central series) (Normal series ) (Subnormal series) (Soluble groups) (Nilpotent groups) (Polycyclic groups) (Super soluble groups) 7

8 מאאאא א 8

9 / / Introduction to group theory W LEDERMANN // 9

10 , H, x, y, z, x H H, H H : H x N H Z y xy H H H H x H Aut H H H,,, x, y N g A Q x, y N g S H K 10

11 SL2,3 2 C Q, S A, B A, B H H H H t 1mod p p t r 1mod p p r H H Tn, F φp g g a b q a τ b a a q N H a N 1 ΦP 1 N H N a 11

12 א א א ROUPS SERIES : : ( ), :, AB ab: a A, b B : A AB ab ab: b B : B AB Ab ab: a A : ( ) H H : gh Hg, g : ( ) : N an; a, N 12

13 : anbn abn, N N N x x N xn : ( ) g g : ( ) : : :, :, : K H K H 13

14 :() : H H : ( ) H H : H : H : ( ) A A N A x ; A A, A x : : ( ) : Z x ; s s ; s : ( ) p p 18 n 0 p 14

15 : ( ) p P p p P p p : ( ) H Aut (Characteristic subgroup) : : : 1 15

16 12 Kurnosenko,NM,On facterisations of finite groups by supersoluble and nilpotent subgroups, problems in algebra,12, Legchekov H V Criterions of supersolubility of some finite factorizable groups, J Algebra and Discrete math 3, Milne J S, roup theory Amer Math Soc,University of Michigan Robinson D A, Course in the theory of roups, Springer Verlag New York Stein Elementary number theory, A computational Approach, Springer Verlag New York Wielandt, H Finite permutation groups,translated from the erman by R Bercov, Academic, New York, Курош АГ Теория груп 3- изд Наука Москва 1967 г 648 с 19 Шеметков ЛА Формации конечных групп изд Наука Москва 1978 г 271 с 78

17 Study about supersoluble groups Abstract The subject of this thesis is the supersoluble groups, the target of this research is to find certain conditions which are applied on any group to become supersoluble The thesis consists of introduction and three section and conclusion and list of references we studied in first section composition, subnormal, normal series and the Commutators, and we mentioned some of examples on which of them We studied in second section the conception of soluble, polycyclic, and nilpotent groups As we studies in second section the intersection between supersoluble groups in the first side and soluble, polycyclic and nilpotent groups in the second ie, we showed that the supersoluble group is soluble group, but the reflection is not true And the polycyclic group is soluble group, but the reflection is not true And so the nilpotent group is soluble group, but the reflection is not true We have reached throw our studies to this research in the third section to the important theorems And we found new classification for supersoluble group which is deferent from last classification As we mentioned the essential theorem which contain three conditions that applied on groups to become supersoluble group Then we have finished the thesis with list of references throw the thesis 79

18 During the preparation of thesis we do the following: 1-Participating in the first scientific conference of mathematics which held in Albaath University in the period between 14-16\10\ we have also translated the following book: Introduction to group theory 3-we have discussed to magazine of Albaath the research classification of supersoluble group 80