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=S5 of permutations on 5 elements.
b) The group G=A5 of even permutations on 5 elements.
c) The group G=GL2(Fp) of invertible two by two
matrices with entries in the field Fp, where p is a prime.
d) The group G=SL2(Fp) of two by two
matrices with determinant one,
with entries in Fp.
2. Let G=GLn(Fp) 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 X1 and X2 are disjoint G-sets and
X is their union, show that fct(X,F) is isomorphic to the
direct sum of fct(X1,F) and fct(X2,F).
b) Show that fct(X1 x X2,F)
is isomorphic to the tensor product (over F) of
fct(X1,F) and fct(X2,F).
4. Let G be a finite group. If V is a
representation for G over F, let chiV 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
chiV(g) = n or -n is a normal subgroup of G.
5. If V is the tensor product of V1 and V2, show that
chiV(g) = chiV1(g)
chiV2(g),
for all g in G.
6. Let X={1,2,3} viewed as an
S3-set, and let V be the vector space of F valued functions
on X, viewed as an S3-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 X1 and X2 are
isomorphic (as G-sets), then the associated F-linear representations
Vj= fct(Xj,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).