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

## Assignment 2

## Due: Monday, October 2.

1. For each of the following groups,
write down its center and give
a list of its non-trivial
conjugacy classes together with the number
of elements in each conjugacy class.

a) The group G=S_{5} of permutations on 5 elements.

b) The group G=A_{5} of even permutations on 5 elements.

c) The group G=GL_{2}(F_{p}) of invertible two by two
matrices with entries in the field F_{p}, where p is a prime.

d) The group G=SL_{2}(F_{p}) of two by two
matrices with determinant one,
with entries in F_{p}.

2. Let G=GL_{n}(F_{p}) be the group of invertible
n by n matrices with entries in the field with p elements.

a) Give an example of a Sylow p-subgroup of G.

b) Give a direct proof, without using the Sylow theorems, that all the
Sylow p-subgroups of G are conjugate. (Hint: use some linear
algebra!)

c) How many distinct Sylow p-subgroups does G contain?

3. If X is a G-set, and F a field, denote
by fct(X,F) the vector space of F-valued functions
on X viewed as a representation for G over F.

a) Show that the assignment which to an object X of the category of G-sets
associates the F-linear representation fct(X,F) gives rise to a functor
from the category of G-sets to the category of F-linear representations of G.

b) If X_{1} and X_{2} are disjoint G-sets and
X is their union, show that fct(X,F) is isomorphic to the
direct sum of fct(X_{1},F) and fct(X_{2},F).

c) Show that fct(X_{1} x X_{2},F)
is isomorphic to the tensor product (over F) of
fct(X_{1},F) and fct(X_{2},F).

4. Let G be a **finite** group. If V is a
representation for G over F, let chi_{V} denote the
character attached to V. If V is n-dimensional over F,
and F is of characteristic zero, show that the set of g in G such that
chi_{V}(g) = n or -n is a normal subgroup of G.

5. If V is the tensor product of V_{1} and V_{2}, show that
chi_{V}(g) = chi_{V1}(g)
chi_{V2}(g),
for all g in G.

6. Let X={1,2,3} viewed as an
S_{3}-set, and let V be the vector space of F valued functions
on X, viewed as an S_{3}-representation.

a) If the characteristic of F is different from 3, show that V is
isomorphic to the direct sum of
two irreducible representations.

b) If the characteristic of F is equal to 3, show that V is not
isomorphic to a direct sum of irreducible representations.

7. Let p be a prime, let G be a group of order a power of p, and let
F be a field of characteristic p. Show that the only irreducible representation
of G over F is the trivial representation.

8. Prove that if two G-sets X_{1} and X_{2} are
isomorphic (as G-sets), then the associated F-linear representations
V_{j}= fct(X_{j},F) are isomorphic as G-representations.
Is the converse true? If so, prove it, otherwise, find a counterexample.

9. Let G be the permutation group S_{n} on n elements.
If g is an element of G of order n, show that g is conjugate to g^{t},
for any integer t which is relatively prime to n.
Use this to conclude that
all the characters of G take rational values.

10. Let p be an odd prime, let F be the finite field with p elements, and let
G = SL_{2}(F) be the group of 2 x 2 matrices with entries in F and with determinant 1.
If g is an element of G of order n prime to p, show that g is conjugate to g^{t}
if and only if t is congruent to either 1 or -1 modulo n.
Use this to conclude that for all characters x of G, the value
x(g) is a real number.
If g is an element of G of order p, show that g is conjugate to g^{t}
if and only if t is a square modulo p.
Use this to conclude that for all characters x of G, the value
x(g) belongs to the field generated
over the rationals
by the square root of p, if p is congruent
to 1 modulo 4, and that it belongs to the field generated by the square root of -p if p is congruent to 3 modulo 4.