18.704 Problem Set 4 Solutions
1. S3 has the following 6 subgroups:
cfw_e, cfw_e, (12), cfw_e, (13), cfw_e, (23), cfw_e, (123), (132), S3.
Here are their character tables. (The three groups of size 2 all have the
same character tables.) Write H = cfw_e, (
How to get character tables of symmetric groups
March 6, 2006
These are some notes on how to construct character tables of symmetric
groups. The focus isnt going to be on why it works, but rather how .
Recall that the conjugacy classes of the symmetric gr
Inner products of induced representations
April 3, 2006
Here is some motivation.
In a previous lecture, we found the character table for S4 by starting with
the following characters (corresponding to partitions) and reducing it.
4
3,1
2,2
2,1,1
1,1,1,1
1,
18.704 Problem Set 3 Solutions
1. G is an abelian group, with identity element 1 (the character of the trivial
representation), as follows. For all g G we have the following.
(1 )(g ) = 1(g )(g ) = 1(g ) = (g ), so 1 = .
[1 (2 3 )](g ) = 1 (g )2 (g )3 (
18.704 Problem Set 2 Solutions
1. Proposition 4 states that if : G GL(V ) is an irreducible representation, any linear transformation T : V V such that (g )T = T (g ) is a
homothety, i.e. T = I for some C. This is fails for representations
over R. The rea
18.704 Problem Set 1 Solutions
1. The subgroups of Q8 are:
cfw_1
cfw_1, 1
cfw_1, i, 1, i
cfw_1, j, 1, j
cfw_1, k, 1, k
Q8
The commutator subgroup contains the element
[i, j ] = iji1 j 1 = ij (i)(j ) = (ij )(ij ) = k 2 = 1.
Similarly [j, k ] = 1 and [k,
18.704 Problem Set 2
Due Friday, Mar. 3, at 3pm in 2-171
1. So far in the course, weve only been considering representations over the
complex numbers C, but we could equally consider real representations,
which are given by homomorphisms from G to GLn (R)
18.704 Problem Set 3
Due Friday, Mar. 17, at
3pm in 2-171
At least one of your answers must be typeset in TEX.
1. Serre, Exercise 3.3.
2. Compute the character table of the following group of order 20.
x, y | x5 = y 4 = e, yxy 1 = x2
3. Suppose a group G
18.704 Problem Set 4
Due Tuesday, April 11, at
3pm in 2-171
At least one of your answers must be typeset in TEX.
(You dont need to submit the TEX code.)
This week, you have a choice.
EITHER do questions 1 and 2, OR do question 3.
(Or do everything if you
18.704 Problem Set 5
Due Friday, May 12, at
3pm in 2-171
At least one of your answers must be typeset in TEX.
(You dont need to submit the TEX code.)
1. Suppose = (1 , 2 , . . . , r ) is a partition, and = t is its conjugate
partition. Show that the only
18.704 Problem Set 1
Due Friday, Feb. 17, at 3pm in 2-171
1. The quaternion group Q8 has the eight elements
1, 1, i, i, j, j, k, k,
and multiplication is given by the rules
i2 = j 2 = k 2 = 1,
ij = ji = k,
jk = kj = i,
ki = ik = j.
Find all the subgroups