# 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.

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 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).

c) 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 G be the permutation group Sn on n elements. If g is an element of G of order n, show that g is conjugate to gt, 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 = SL2(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 gt 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 gt 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.