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

## Assignment 3

## Due: Monday, October 16.

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

**3.** 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 Alt^{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.

**4.** 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.)

**5.**
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.)

**6.**
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}.)

**7.**
Show that G has two distinct conjugacy classes
of elements of order 7 (to be denoted 7A and 7B),
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.)

**8.** 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.

**9.** 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.

**10.** Let W_{4} be the alternating square
of the representation V_{2} of exercise 8.
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.

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