Math 551, Assignment 1

Due Thursday, Sept. 14 in class

1. If X and Y are two sets, finite or infinite, we say that |X| = |Y | if and only if there exists a bijection between X and Y. Show that if |X| = |Y |, then Σ_{X} ≅ Σ_{Y}.

2. Let the finite group G act on the finite set Ω. For each g ∈ G let θ(g) = |F ix_{Ω}(g)|. Show that ∑_{g∈G} θ(g) = k|G|, where k is the number of orbits of G on Ω. (Hint. In the transitive case, count the number of pairs (g, α) such that gα = α, and do the count in two different ways.)

3. Suppose that G acts on Ω. For any integer n, let Ω^{n} = Ω × ⋯ × Ω (n factors). Define in a natural way an action of G on Ω^{n}. If G = Σ_{Ω}, then how many orbits does G have on Ω^{n} for n = 2, 3, 4? You may assume that Ω is sufficiently large, but do say how large that is.

4. Let G and H be cyclic groups, neither of which is the trivial group (of order 1). Give necessary and sufficient conditions for the direct product G × H to be cyclic.

5. Let G = Z_{n}. Let R = Z/nZ be the ring of integers modulo n, and R^{×} the group of units (multiplicatively invertible elements) of R, under the operation of multiplication. Show that Aut(G) ≅ R^{×}.

6. Show that a group of exponent 2 is necessarily abelian, but construct a non-abelian subgroup of SL_{3}(Z/3Z) which has order 3^{3} and exponent 3.

7. Show that SL_{2}(Z) = ⟨ [ \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ] , [ \begin{matrix}0 & 1 \\ −1 & −1 \end{matrix} ] ⟩. Conclude that P SL_{2}(Z) is generated

by an element of order 2 and an element of order 3. (P SL_{2}(Z) is the group of transformations of C ∪ {∞} of the form z ↦ (az + b)/(cz + d) with a, b, c, d ∈ Z and ad − bc = 1.)

8. Show that if a group G is generated by two elements g, h of order 2, then G has a cyclic subgroup H such that |G : H| = 2 and g, h ∉ H. What are the possible isomorphism types for G? Show that P SL_{2}(Z) is not generated by two elements of order 2.