University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (August-2004)
Point of Post: This is the August 2004 part of the post started here.
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.
Algebra Qual(Ph.D. Version)-August 2004
a) Suppose that and . Let and be odd. Show that .
b) Let be a finite group and suppose there exists subgroups
with for . Suppose that has odd order, show that .
c) Let be a group of order where is odd. Suppose that contains a normal subgroup of order . Show that there exists subgroups
with for .
a) Since we know that and thus . Let then , since is odd and we know we know that is odd and so for some . Thus, that said if is the canonical projection onto then and thus by definition . Since was arbitrary the conclusion follows.
b) By definition we have that of course and since and is odd we have by a) that . Similarly, since and is odd we may conclude that . Continuing in this way we clearly get that . Since was arbitrary the conclusion follows.
c) By Sylow’s Theorems we know that has a subgroup of order , and by Sylow’s theorems has a subgroup of order , and continuing we get a chain of subgroups
such that . We recall then that since we have that and since so that is trivial we may conclude from a common result that . Similarly, we know that and since so that is trivial we may conclude that . Lastly, noting that since so that we note that
Remark: Of course this proves that every group of order where is odd which has a normal subgroup of order is solvable.
Problem 6: Let be a representation of the finite group . Define by the block matrix
a) Show that is a representation of .
b) Show that the number of times that the trivial representation of occurs in equals the number of times that the trivial representation occurs in the decomposition of .
a) Consider the representation where is the trivial representation on and is the other non-trivial representation on . It’s evident that and so trivially a homomorphism.
b) We need only compute that where the inner product is taken on the group algebras and respectively. That said, from a) we see that where . It’s easy then to see that:
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 6, 2011 - Posted by Alex Youcis | Algebra, 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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