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Morphisms of Fusion Systems
KTH, School of Engineering Sciences (SCI).
2014 (English)Independent thesis Basic level (degree of Bachelor), 10 credits / 15 HE creditsStudent thesis
Abstract [en]

A fusion system on a finite group G with a Sylow p-subgroup P is a category on P with all subgroups of P as objects and group homomorphisms induced by conjugation in G as morphisms and was first introduced by L. Puig around 1990 in order to aid his research in finite group theory. The idea turned out to be fruitful and today, the theory of fusion systems is an active field in mathematics, with applications to topology, representation theory and finite group theory.

In this paper, we will, among other things, see how fusion systems aid in solving problems in finite group theory. We begin with an introduction to the theory with basic examples and then proceed to prove two famous theorems named after Burnside and Frobenius. However, to finish the proof of Frobenius’ theorem, we will require the focal subgroup theorem, whose proof requires transfer theory and is thus discussed. Afterwards, we introduce abstract and saturated fusion systems, in which one disregard the underlying group, and later prove that every fusion system on a finite group is saturated. We end with a discussion of morphisms of fusion systems, utilizing the concept previously developed, and generalize the isomorphism theorems to saturated fusion systems.

The presentation is well adapted for undergraduate students with limited knowledge of group and category theory and no previous knowledge of fusion systems is assumed.

Place, publisher, year, edition, pages
2014. , 26 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-154553OAI: oai:DiVA.org:kth-154553DiVA: diva2:757638
Educational program
Master of Science in Engineering -Engineering Physics
Supervisors
Available from: 2014-10-23 Created: 2014-10-23 Last updated: 2014-10-23Bibliographically approved

Open Access in DiVA

Karl Amundsson kandidatexam(2374 kB)120 downloads
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CiteExportLink to record
Permanent link

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Citation style
  • apa
  • harvard1
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
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