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 uptodate 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
Problem 1:
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 …”)
Proof:
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, .
c) We use the fact that since that is inevitably cyclic and so by a common theorem we must have that is abelian. Then, by the structure theorem we may conclude that either or .
References:
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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