189-570A: Higher Algebra I

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).