**189-570A:** Higher Algebra I

## Assignment 3

## Due: Monday, October 16.

** I have decided to divide the class into
two groups, group A and group B. Group A is for the
people who have a somewhat stronger
background and are not struggling too much with the assignments.
Group B is for the students with somewhat less preparation.
**

Indicate clearly at the start of your assignment
what group you place yourself in.

If you place yourself in group B, you are
required to do only problems
1 through 5. However, at the end of the term there will be a price to
pay for this lighter assignment load: most likely I will
ask you to write a project or a take-home assignment developping in more
depth some aspect of the course. The
idea, for people in Group B, is to spread the work out more evenly
over the semeser so that you are not handicapped by
your less strong preparation.

**1**. Recall that for an element g of a group G, the
centralizer of g in G - the group of elements
of G which commute with g - is denoted by Z(g).

a.
Show that the function which to g in G associates the cardinality of
Z(g) is the character of a (naturally occuring) representation of G.
What is this representation?

b. Using part a, compute the average
over G of the cardinality of Z(g) -- the average size of the
centralizer of an element of G.

**2.** Let G = { 1, -1, i, -i, j, -j, k, -k}
be the quaternion group of order 8.
(Recall that the multiplication of quaternions is defined by the rules
ij = -ji = k, jk = -kj = i, ki = -ik = j.)

a. Show that G is a non-abelian group which is not isomorphic
to the dihedral group of order 8.

b. Compute the character table of G and compare it with the character table for
D_{8} computed in class. What do you observe?

**3.** Compute the character table for S_{5}, the symmetric group
on 5 elements.

**4.** Let V be a representation for G, and let
W=V*V be the tensor product of V with itself.

a.
The symmetric square of V, denoted Sym^{2}(V), is the
subspace of W spanned by expressions of the form
v_{1}*v_{2}+v_{2}*v_{1}.
Show that the character X attached to Sym^{2}(V)
is given by the formula

X(g) = (chi(g)^{2}+chi(g^{2}))/2

where chi denotes the character attached to V.

b.
The alternating square of V, denoted Alt^{2}(V), is the
subspace of W spanned by expressions of the form
v_{1}*v_{2}-v_{2}*v_{1}.
Show that the character X attached to Sym^{2}(V)
is given by the formula

X(g) = (chi(g)^{2}-chi(g^{2}))/2

where chi denotes the character attached to V.

Let G = GL_{3}(F_{2}) be the
group of invertible 3 x 3 matrices with
entries in the field with two elements.
It acts naturally by linear transformations on the
space W of column vectors of size 3 with entries in F_{2}.

*The goal of the following series of exercises is
to partially calculate the character table for G. *

Note that you are not obliged to
follow the hints.

**5.** Show that G has a unique conjugacy class (to be denoted 2A) of elements of
order 2, which is of cardinality 21.
(Hint:
If T is an element of order 2 in G, show that T has minimal
polynomial (T-1)^{2}. Conclude that the linear endomorphism
U=(T-1) has a two dimensional kernel and that the image of U is a
one-dimensional subspace of Ker(U). Show that the datum
(Image(U) , kernel(U)) determines T completely.)

**6.**
Show that G has a unique conjugacy class (to be denoted 4A) of elements of
order 4, which is of cardinality 42.
(Hint: Show that the 21 distinct Sylow 2-subgoups of G are isomorphic to
D_{8} and that the intersection of any two
of them is a group of exponent 2.)

**7.**
Show that G has a unique conjugacy class (to be denoted 3A) of elements of
order 3, which is of cardinality 56.
(Hint: Show that the characteristic polynomial of an
element T of order 3 is equal to
(x+1)(x^{2}+x+1) and that T is completely determined by the following
data: the kernel W_{1}
of T^{2}+T+1 and the kernel W_{2} of T+1,
yielding a decomposition of W into a direct sum of a two-dimensional
and a one dimensional
subspace,
together with the extra datum of a cyclic permutation of order 3
on the non-zero vectors in W_{1}.)

**8.**
Show that G has two distinct conjugacy classes
of elements of order 7 (to be denoted 7A and 7B) of elements of
order 7, corresponding to the linear transformations having
characteristic polynomial
x^{3}+x^{2}+1 and x^{3}+x+1 respectively.
(Hint: Show that G has 8 distinct Sylow 7-subgroups and hence
48 elements of order 7.)

**9.** Let X be the set of non-zero vectors of W and let
V_{2}=fct^{0}(X,C) be the set of complex-valued
functions on X of sum 0. Compute the character attached to V and
show that V is an irreducible 6-dimensional representation
of G.

**10.** Let X be the set of Sylow 7-subgroups of G,
on which G acts by conjugation, and let
V_{3}=fct^{0}(X,C) be the set of complex-valued
functions on X of sum 0. Compute the character attached to V and
show that V is an irreducible 7-dimensional representation
of G.

**11.** Let W_{4} be the alternating square
of the representation V_{2} of exercise 9.
Show that V_{4} decomposes as a direct sum of
V_{3} and a new irreducible representation V_{4}
of dimension 8. Write down the character of V_{4}.
From the work you have done so far, show that the 6 irreducible representations of G are of dimension 1,6,7,8, 3 and 3 respectively.
Of these, only the characters of the
two irreducible three-dimensional representations have
not been computed.

**12.** (optional) Write down the character for the 6 dimensional
representation which is the direct sum of the two irreducible
three-dimensional representations attached to G.