6627

6178

كارشناسي ارشد

جبر

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1390

[هفت]، 78ص

ص.ع. به فارسي و انگليسي

1396/10/12

كتابنامه

علوم رياضي

رياضي

ID6178

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

On the Number of Centralizers in Finite Groups Abdollah Sa yan Boldaji A sa yan boldaji iut ac ir June 2011 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr B Taeri b taeri@cc iut ac ir2011 MSC 20F99 20D60 20D99 Key words centralizer centre of G n centralizer primitive n centralizer cover projective special lineargroup Suzuki group Abstract In this thesis investigate number of centralizers in nite groups Let G be a group andCent G denote the set of centralizers of single elements of G Let n 1 be an integer thatis Cent G CG g g G where CG g is the centralizer of the element g in G A groupG is called n centralizer if Cent G n and primitive n centralizer if Cent G Z G Cent G n where Z G denotes the centre of G In this thesis for a nite group G wegive some interesting relations between Cent G and the maximum number of the pairwisenon commuting elements in G Also we study all n centralizer nite groups for n 7 and 8 Using these results we see that there is no nite group G with the property that Cent G Cent G Z G 8 In this thesis all groups will be nite and we use the usual notation for example An Sn P SL n q and Sz q denote the alternating group on n letters thesymmetric group on n letters the projective special linear group of degree n over the nite eld of size q and the Suzuki group over the eld with q elements respectively Cn denotesthe cyclic group of order n D2n stands for dihedral group of order 2n A subgroup H of Gis called a proper centralizer of G if H CG x for some x G Z G It is clear that agroup is 1 centralizer if and only if it is abelian Belcastro and Sherman have the followingresults i There is no 2 centralizer and no 3 centralizer group ii A nite group G is4 centralizer if and only if G Z G C2 C2 iii A nite group G is 5 centralizer if andonly if G Z G C3 C3 or S3 Using these results it is easy to see that there is no niteprimitive 4 centralizer group and a nite group G is primitive 5 centralizer if and only ifG Z G S3 and has shown that if G is a nite 6 centralizer group then G Z G D8 orA4 or C2 C2 C2 or C2 C2 C2 C2 We show where G is a 7 centralizer group if and only ifG Z G C5 C5 or D10 or x y x5 y 4 1 y 1 xy x3 Let G be a nite 8 centralizergroup Then G Z G C2 C2 C2 or A4 or D12 and There is no nite primitive 8 centralizer group we study the number Cent G for all minimal simple groups Recall thatA minimal simple group is a non abelian simple group all of whose proper subgroups aresolvable Using these results we see that there exist nite simple groups G and H with theproperty that Cent G Cent H but G H This result gives a negative answer to aquestion raised by A Ashra and B Taeri We also investigate all nite semi simple groupsG with Cent G 73 Recall that A nite group is called semisimple if it has no nontrivialnormal abelian subgroups A Ashra and B Taeri have investigated the structure of nitegroups with at most 21 element centralizers and also using the classi cation of nite simplegroups they proved if G is a nite simple group and Cent G 22 then G A5 1

بيزن طائري

محمد مسكوري

محمدرضا ريسمانچيان، محمدرضا ودادي