SOLUTIONS FOR ASSIGNMENT 1, M6390 FALL 08
(1) Group homomorphisms were dened in the notes as structure-preserving
maps. However, the denition contains a redundancy. Prove that the
multiplication-preserving property of a group homomorphism implies the
foll
SOLUTIONS FOR ASSIGNMENT 2, M6390 FALL 08
(1) Serres book, problems 2.1, 2,2, 2.4.
Solution to 2.1:
One way to do this problem is to use Proposition 3 in the book, which
makes the problem an easy algebra exercise. However, its not too
hard to do it direc
SOLUTIONS FOR ASSIGNMENT 3, M6390 FALL 08
(1) Solution to 3.1
Let : G GL(V ) be a representation, with G an abelian group. Since
G is abelian, for all g, h G, we have g h = h g . This makes each g a
G-equivariant map from V to V . By Schurs lemma, each g
SOLUTIONS FOR ASSIGNMENT 4, M6390 FALL 08
3.4 This problem is an exercise in applying the examples in section 3.3 of Serre.
The regular representation G of G is induced by the regular representation
H of H . Decompose H into a direct sum of irreducible re
SOLUTIONS FOR ASSIGNMENT 5, M6390 FALL 08
Q8 has ve conjugacy classes. Thus there are ve irreps. The sum of the squares
of their dimensions must add up to eight. The only possibility then is that four of
the representations have degree one and one has deg
REPRESENTATIONS OF FINITE GROUPS AND APPLICATIONS
MATH 6390, FALL 2008
COURSE NOTES
TOBIAS HAGGE
1. Introduction to Groups
Denition. A binary operation on a set S is a function : S S S , one writes
a b to denote (a, b).
Denition. A binary operation on S i
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