189-570A: Higher Algebra Assignment 1

Due: Monday, October 2

1. Read through Chapter 1 of Lang’s book. The goal here is to familiarise yourself with the basic definitions and results that we did not discuss in class. The material here is sometimes dry, but not terribly difficult.

2. Show that the intersection of two finite index subgroups of a group G is also of finite index.

3. Let G be a non-abelian finite simple group, and let n be an integer satisfying 1 < n! < #G. Show that G contains no conjugacy class of size n. If p is a prime dividing the order of G, show that the number of Sylow p-subgroups of G is > n.

4. Write down the class equations for the symmetric group S_{6} and alternating group A_{6}, and show that A_{6} is a simple group.

5. Let G be a subgroup of A_{n} of index n. Show that G is abstractly isomorphic to A_{n−1}.

6. Let G be a simple group of cardinality 60. Show that G has exactly 6 Sylow 5-subgroups. Conclude that G is isomorphic to a subgroup of A_{6}. Using exercise 5, conclude that G is isomorphic to A_{5}.

7. Let G be a finite group and let X be a G-set consisting of exactly t orbits. Given g ∈ G, let FP_{X}(g) be the number of fixed points of g acting on X:

FP_{X}(g) := #{x ∈ X such that gx = x.}.

Show that

\frac{1}{#G} ∑_{g∈G} FP_{X}(g) = t.

8. We keep the notations of exercise 7. The action of G on X is said to be two-fold transitive if for all x, y, x′, y′ ∈ X with x ≠ y and x′ ≠ y′, there exists g ∈ G with gx = x′ and gy = y′. Show that this propery holds if and only if

∑_{g∈G} FP_{X}(g)^{2} = 2(#G).

9. Let G = GL_{3}(F_{2}) be the group of invertible 3 × 3 linear transformation with entries in the field F_{2} with two elements. What is the cardinality of G? Show that G is simple.

10. We retain the notations of problem 9. Let X_{1} be the set of non-zero vectors in F_{2}^{3}, and let X_{2} be the set of two-dimensional subspaces of F_{2}^{3}. Both sets are equipped with their natural structures of G-sets (using the natural action of G on F_{2}^{3} by linear transformations.) Prove the following assertions made by J-P. Serre in his seminar lecture on Thursday.

a) For all g ∈ G, we have FP_{X_{1}}(g) = FP_{X_{2}}(g).

b) The sets X_{1} and X_{2} are not isomorphic as G-sets