Study On The Sylow Decomposition And Antihomomorphism Of Some Finite Groups

In numerous group theory research,the study of finite group from theory to practical application are occupies the important position.It also became the most active research in recent decades a branch,through the efforts of many mathematicians,made in the research field of finite group has made a series of breakthrough,and ultimately makes the classification problem of the one group gets a complete solution.This is a breakthrough of it is not easy to,so the research on the classification of the general group is harder,makes the understanding of structure is especially important,this thesis is based on the analysis and research the related properties of subgroup local and special structure,to identify the structure of the group itself and quality.In isomorphic sense,the properties and structures of subgroups are obtained by studying the direct product decomposition,so as to determine the structure and properties of groups themselves.Decomposition method is particularly important at this moment,we know that when the group is solvable group,the group can be broken down to the west of the rocky bottom of product.Because of the west of the rocky bottom decomposition is a group of necessary and sufficient conditions for solvability,so want to find a finite group can break down for the subgroup of the general method,makes the solvable group of west of the rocky bottom decomposition method is a special case of the general decomposition method,then introduced the concept of synthetic block,made the problem solution are obtained.A good decomposition approach provides guarantee for the smooth completion of this paper.The first chapter of this paper introduces the basic definitions and lemmas related to this paper.The second chapter uses the theorem of Sylow to break down some of the finite groups and uses a group watch to make a representation of the group,and it’s a breakdown of a few specific groups:1.With respect to the dissociation and group table representation of the class 6 groups,there are two cases for the 6 order group G decomposition:G contains one 2-Sylow subgroup,one 3-Sylow subgroup;G contains three 2-Sylow subgroup,a third-order subgroup.2.Dissociation and group table representation of order 10 groups.There are two cases for the group G decomposition.G contains one class 2 cycle group,and one fifth cycle group.G contains five second order groups.3.In terms of the decomposition and group table of the 12th order group,there are three cases of group G decomposition of the 12th order group:G contains a third order subgroup and a fourth order subgroup;G contains one third order subgroup,three fourth-order subgroups;G contains one fourth order subgroup,four third-order subgroups.4.The breakdown of the 35-order group and the group form indicates that there is a situation where the 35 group of G is broken down,and there is a fifth cycle group,a seventh cycle group.5.The decomposition of the 18th order group.There are three scenarios for group G decomposition of the 18th order group:G contains a 2-Sylow subgroup and a 3-Sylow subgroup;G contains three 2-Sylow subgroups,a 3-Sylow subgroup;G contains nine 2-Sylow subgroups,one 3-Sylow.6.The decomposition of the 20th order group.G contains one 2-Sylow subgroup,one 3-Sylow subgroups;G contains five 2-Sylow subgroup,one 5-Sylow subgroups;7.The decomposition of the 255th order group.There’s one for the 255 order group,and it’s got a loop group of C,,and the cycle is C5,and the cycle group is C17Chapter three is A theory about T.Asai and T.Yoshida,which is based on the structure of the loop group and the characteristics of the group of elements,and using the basic methods of algebra and number theory,we have calculated the number of antihomomorphisms in the two cycles,namely |Hom反(Cm,Cn)| =(m,n),and then we have more questions about the discussion.which is the idea of T.Asai and T.Yoshida.