University of Maryland College Park Qualifying Exams (Group Theory and Representation Theory) (August2004)
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 uptodate PDF of the parts of the exam I’ve finished.
Algebra Qual(Ph.D. Version)August 2004
Problem 1:
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 .
Proof:
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
and
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 .
Proof:
a) Consider the representation where is the trivial representation on and is the other nontrivial 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:
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 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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(Q,+) has uncoutable sub groupes
Comment by mohamad  May 13, 2011 
Indeed friend, it does. What does that have to do with this post though?
Comment by Alex Youcis  May 14, 2011 
Lol, mohamad is confused.
Nice post!
Comment by Bruno Joyal  September 5, 2011