MA3E1 Groups and Representations
Assignment 4
Deadline: Friday 30 November 2012, 3pm
Write your name and student number on your solution sheet. Mention
your department if it is not mathematics.
(4.1) Let be an irreducible representation of a nite group G.
Test MA3E1 Groups and Reps
Version 1
Monday 12 November 2012
Name:
Student Number:
This test covers the material lectured up to and including theorem 107 (orthogonality of characters, page 36). You can bring any notes you like, but no electronic gadgets o
Test MA3E1 Groups and Reps
Version 1
Name:
Tuesday 16 November 2010, 12pm
Department:
Student Number:
This test covers the material lectured up to and including chapter 5. You can bring any notes you like,
but no electronic gadgets or library books.
Every
MA 3E10
THE UNIVERSITY OF WARWICK
THIRD YEAR EXAMINATION: June 2010
MA3E1 GROUPS AND REPRESENTATIONS
Time Allowed: 3 hours
Read carefully the instructions on the answer book and make sure that the particulars
required are entered on each answer book.
Calc
MA 3E10 GROUPS AND REPRESENTATIONS
THE UNIVERSITY OF WARWICK
THIRD YEAR EXAMINATION: April 2011
MA3E1 GROUPS AND REPRESENTATIONS
Time Allowed: 3 hours
Read carefully the instructions on the answer book and make sure that the particulars
required are enter
MA 3E10 GROUPS AND REPRESENTATIONS
THE UNIVERSITY OF WARWICK
THIRD YEAR EXAMINATION: April 2012
MA3E1 GROUPS AND REPRESENTATIONS
Time Allowed: 3 hours
Read carefully the instructions on the answer book and make sure that the particulars
required are enter
MA3E1 Groups and Representations
3
1 Groups
1.1
Groups
Denition 1. A group consists of a set G and a binary operation G G G :
( x, y) xy such that:
Associativity: We have x( yz)
= ( xy) z for all x, y, z G.
G such that 1x = x = x1 for all x G.
We call 1
MA3E1 Groups and Representations
51
(7.7) Let be a character of an innite group G. Prove that g ( g1 ) and
g ( g) are again characters of G. Give an example where ( g1 ) = ( g).
(7.8) Let i (1 i k) be the irreducible characters of a nite group G. Let C j
26
Chapter 4
September 20, 2012
Denition 81. A CG-module is said to be simple if it is nonzero and it has no
submodules other than 0 and itself.
Every 1-dimensional CG-module is simple.
Internal direct sums. Let V be a vector space and X, Y V linear subsp
MA3E1 Groups and Representations
Assignment 2
Deadline: Tuesday 6 November 2012, 3pm
Write your name and student number on your solution sheet. Mention
your department if it is not mathematics.
(2.1) (Adopted from the 2011 exam.) Put G = x, y | x2 , y3 ,
MA3E1 Groups and Representations
Assignment 1
Deadline: Friday 19 October 2012, 3pm
Write your name and student number on your solution sheet. Mention
your department if it is not mathematics.
(1.1) We say that a square matrix X is upper triangular, if al
MA3E1 Groups and Representations
Assignment 3
Deadline: Tuesday 20 November 2012, 3pm
Write your name and student number on your solution sheet. Mention
your department if it is not mathematics.
(3.1) Let , be irreducible representations of a nite group G
Test MA3E1 Groups and Reps
Version 1 Wednesday 24 February 2010
Name:
Student Number:
This test covers the material lectured up to and including chapter 5. You can bring any notes you
like, but no electronic gadgets or library books. Marks are given as fo