MA32055: Groups and their representations
[Page last updated: 22 May 2025]
| Academic Year: | 2025/26 |
| Owning Department/School: | Department of Mathematical Sciences |
| Credits: | 10 [equivalent to 20 CATS credits] |
| Notional Study Hours: | 200 |
| Level: | Honours (FHEQ level 6) |
| Period: |
- Semester 2
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| Assessment Summary: | EXCB 100% |
| Assessment Detail: |
- Closed-book written examination (EXCB 100%)
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| Supplementary Assessment: |
- Like-for-like reassessment (where allowed by programme regulations)
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| Requisites: |
Before taking this module you must take MA22017 OR take MA20217
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| Learning Outcomes: |
- Demonstrate knowledge and understanding of the basic theory of groups and their representations.
- Work with Sylow Theory and apply this to show that groups of certain orders must be simple.
- Work with the structure theorem of abelian groups in theory and practice.
- State and prove the main theorems of Maschke and Schur and their applications in representation theory and character theory.
- Apply these results to problems in Group Theory.
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| Synopsis: | You will explore one of the main branches of Algebra, Group Theory, that can be seen as the mathematical theory that deals with symmetry.
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| Content: | Semidirect products, finite nilpotent groups and their characterisations, the Frattini subgroup and the Burnside Basis Theorem, solvable groups, some Sylow Theory and material on simple groups.
Group algebras, their modules and associated representations. Maschke's theorem and complete reducibility. Irreducible representations and Schur's lemma. Decomposition of the regular representation. Character theory and orthogonality theorems.
Further topics may include: Burnside's (p,q)-Theorem, Burnside's normal p-complement theorem, Finite p-groups, character theory for abelian groups, Iwasawa's lemma and the simplicity e.g. of PSL2(Fp), Induced representations and Frobenius reciprocity.
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| Course availability: |
MA32055 is Optional on the following courses:
Department of Computer Science
- USCM-AFB32 : BSc(Hons) Computer Science and Mathematics (Year 3)
- USCM-AAB20 : BSc(Hons) Computer Science and Mathematics with Study year abroad (Year 4)
- USCM-AKB20 : BSc(Hons) Computer Science and Mathematics with Year long work placement (Year 4)
- USCM-AFM32 : MComp(Hons) Computer Science and Mathematics (Year 3)
- USCM-AKM32 : MComp(Hons) Computer Science and Mathematics with professional placement (Year 3)
- USCM-AKM32 : MComp(Hons) Computer Science and Mathematics with study abroad (Year 3)
Department of Economics
- UHES-AFB12 : BSc(Hons) Economics and Mathematics (Year 3)
- UHES-AAB04 : BSc(Hons) Economics and Mathematics with Study year abroad (Year 4)
- UHES-AKB04 : BSc(Hons) Economics and Mathematics with Year long work placement (Year 4)
- UHES-ACB04 : BSc(Hons) Economics and Mathematics with Combined Placement and Study Abroad (Year 4)
Department of Mathematical Sciences
- USMA-AAB16 : BSc(Hons) Mathematical Sciences with Study year abroad (Year 4)
- USMA-AKB16 : BSc(Hons) Mathematical Sciences with Year long work placement (Year 4)
- USMA-AFB30 : BSc(Hons) Mathematics (Year 3)
- USMA-AAB14 : BSc(Hons) Mathematics with Study year abroad (Year 4)
- USMA-AKB14 : BSc(Hons) Mathematics with Year long work placement (Year 4)
- USMA-AAB02 : BSc(Hons) Mathematics and Statistics with Study year abroad (Year 4)
- USMA-AKB02 : BSc(Hons) Mathematics and Statistics with Year long work placement (Year 4)
- USMA-AFB33 : BSc(Hons) Mathematics, Statistics and Data Science (Year 3)
- USMA-AAB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Study year abroad (Year 4)
- USMA-AKB20 : BSc(Hons) Mathematics, Statistics, and Data Science with Industrial Placement (Year 4)
- USMA-AFM30 : MMath(Hons) Mathematics (Year 3)
Department of Physics
- USPH-AFB26 : BSc(Hons) Mathematics and Physics (Year 3)
- USXX-AAB04 : BSc(Hons) Mathematics and Physics with Study year abroad (Year 4)
- USXX-AKB04 : BSc(Hons) Mathematics and Physics with Year long work placement (Year 4)
- USPH-AFM26 : MSci(Hons) Mathematics and Physics (Year 3)
- USXX-AAM01 : MSci(Hons) Mathematics and Physics with Study year abroad (Year 4)
- USXX-AKM01 : MSci(Hons) Mathematics and Physics with Year long work placement (Year 4)
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Notes: - This unit catalogue is applicable for the 2025/26 academic year only. Students continuing their studies into 2026/27 and beyond should not assume that this unit will be available in future years in the format displayed here for 2025/26.
- 好色tv and units are subject to change in accordance with normal University procedures.
- Availability of units will be subject to constraints such as staff availability, minimum and maximum group sizes, and timetabling factors as well as a student's ability to meet any pre-requisite rules.
- Find out more about these and other important University terms and conditions here.
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