Expansioins of C-Sets and D-Sets Having Jordan Automorphism Groups and Some Related Questions

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6.10 Colinality of Aut(w, D)

\Ji as a ^diS ·

··

JOmt ^union

U

^{E(O .} ^.

128 b^eth^einduced C . , i) of moieties E(O i) .

-r^elation on E( . . , , ⁱ E / at th '

h

E Hi \H�(O,i) d 0, i) mduced b D e node 0. Let C·

i an l^et f b h y on w B ⁱ

i E J Th

e t e ^el^ement f . ·or each i

I

· ^en cl^early f d

U _·

⁰ G which · d E choose

�

H·

^· ^{m uces J}

proves our th

i contrary to o . i on E(O i) f

eorem. •

m assumpt· , or each

ion and th. ts contradiction

Chapter 7 Conclusion

In this final chapter we give a b^rief ^summa^ry of the w^ork that has b^een presented

wor

in ^this thesi^s. We end by ^statiⁿg ^some ôpen p^roblems clo^seto ôu^rarêa of k

7.1 A brief s ummary of the w ork

We cons^truc^ted a C-set fl and studie^dthêautômo^rphîsm g^roup ôf the C-set.

We have shôwn ^that aⁿy elemeⁿt of the aûtomorphi^sm g^roup of ^thêC-^set caⁿ be expressed as a prôduct ôf chain autômo^rphîsm^saⁿd b^ranch automôrphîsm^s. We then imposed extra rela^tions '.,, V, Land R ^{on t}he C-^set. W^{e st}u^die^{d s}ome properties of the automôrp^hi^sm grôupsof the C-^se^tfl wit^ht^he êx^tra ^relatioⁿs.

From literature the autômôrphism groûp ôf a C-sêt i^sk^nown to bêa Jôr^daⁿgroup. We have shown that the au^tomo^rphism grôup^sof the ^strûctûrês wi^th th^{e e}x^tra

I . • osed are also Jordan groups. We have determin^ed a minimal Jordan re ations nn P

G

h th t the impôsitîon ôf êxt^ra rela^tiôn ônthe un^derlyⁱⁿg ^st^ructûrê . J ·d autômôrphi^sm g^roup. Fu^rt^her, wêhave ^shown that a group o sue a

no longer admits a o1 an . .

t · s thêclass ôf t^ranslatiôn b^raⁿch aûtomo^rph1Sm^siⁿ branch relation that côn ain

be im^posed oⁿt^heC-^set and ^stⁱll a^dmⁱt a J^or^daⁿ its automorphism g^roupcan

d c^hain ^relatiôn can bê,mp^{osed on t}hêC-set a^nd automorphism g^roUP· An aⁿy

still admit a Jordan autom^orp^hismg^roup. 129 TH-2714_964601

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Expansioins of C-Sets and D-Sets Having Jordan Automorphism Groups and Some Related Questions

U

h

I

U ·

H·

Chapter 7 Conclusion

7.1 A brief s ummary of the w ork

G

U _·