University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) ( January 2003))
Point of Post: A friend of mine who attended UMD CP for his Ph.D. has told me that they are famous for their brutally hard qualifying exams. Thus, to keep sharp and as a challenge I’d like to complete all of the qual exams for Algebra, Analysis, and Topology/Geometry eventually–although this is an ambitious venture. So, I plan to break them up into subsections and do all of the ____ subject (ring, field,…) in the Algebra quals and then maybe do all the ____ subject( measure theory, complex analysis, …) in the Analysis quals etc. Since what I’m currently doing on my blog has been representation theory and group theory mostly I thought I’d start with those two. Thus, this will be the first post in a series to solve the Group Theory/Representation Theory problems on the quals.
NOTE: Just because I am posting references (which are really more like further reading) I will not use any book while taking these.
NOTE: Instead of doing this post by post I thought it would make more sense to have it all consolidated. See here for the up-to-date PDF of the parts of the exam I’ve finished.
Remark: The set of all the algebra exams may be found here.
Algebra Qual(Ph.D. Version)-January 2003
a) Let be a group of order . Show that (Hint: you may assume that the group has order ).
b) Let be a group of order . Show that contains a subgroup of order .
c) Find all groups of order .
(Do not use theorems of the form “every group of order …”)
a) Since we know that or . Thus, it assumes to prove this is true for those two groups. So, first define by
is the canonical projection. Evidently is a vector space over and evidently every group endomorphism will be a linear endomorphism. Thus, since by definition is by definition where is the ordered basis we have from basic linear algebra that and is a monomorphism. Thus and so by Lagrange’s theorem .
For we appeal to the common fact that and so where here is Euler’s totient function.
b) Let denote the number of Sylow -subgroups of . By Sylow’s theorems we must have that and so but since we must conclude that . Thus, by a common theorem we may conclude that if is the unique Sylow -subgroup that . Thus, consider the map by where is the inner automorphism (this is well defined since is normal). This is evidently a homomorphism and so what we have by the first isomorphism theorem is that and so in particular . But, by definition the centralizer of in and so by part a) we may conclude that . But, since we may conclude that is trivial and so . Thus, .
1. Dummit, David Steven., and Richard M. Foote. Abstract Algebra. Hoboken, NJ: Wiley, 2004. Print.
2. Simon, Barry. Representations of Finite and Compact Groups. Providence, RI: American Mathematical Society, 1996. Print.
May 1, 2011 - Posted by Alex Youcis | Algebra, Fun Problems, Group Theory, Representation Theory, UMaryland Qualifying Exams | Algebra, Group Theory, January 2003, Qualifying Exams, Representation Theory, University of Maryland Qualifying Exams
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My name is Alex Youcis. I am currently a senior a first year graduate student at the University of California, Berkeley.
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