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( alois theory) ( 1736 1823) Joseph Louis Lagrange ( 1789 1857 ) Augustine Louis Cauchy Ludwid Sylow ( 1838 1922 ) Camille Jordan Milter Shur Burnside Helpert Frabinous 6
(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
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/ / Introduction to group theory W LEDERMANN // 9
, 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
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
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
: anbn abn, N N N x x N xn : ( ) g g : ( ) : : :, :, : K H K H 13
:() : 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
: ( ) p P p p P p p : ( ) H Aut (Characteristic subgroup) : : : 1 15
12 Kurnosenko,NM,On facterisations of finite groups by supersoluble and nilpotent subgroups, problems in algebra,12,1998 13 Legchekov H V Criterions of supersolubility of some finite factorizable groups, J Algebra and Discrete math 3, 2005 14 Milne J S, roup theory Amer Math Soc,University of Michigan 2007 15 Robinson D A, Course in the theory of roups, Springer Verlag New York 1980 16 Stein Elementary number theory, A computational Approach, Springer Verlag New York 2007 17 Wielandt, H Finite permutation groups,translated from the erman by R Bercov, Academic, New York, 1964 18 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
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
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\2008 2-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