**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 which was
introduced in the
previous asignment
(question 9) where you computed its cardinality.

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) 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).

b) 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 M and N be modules over a ring R.
Show that the direct sum of M and N is projective (resp. injective)
if and only
if both M and N are projective (resp. injective).