MODERN ALGEBRA (MATH 210) PROBLEM SET 4

From the text: 5.27–5.31. If you use results of earlier exercises (see 5.29), make sure to prove them!

1. Suppose F is a finite field. Find the product of the elements of F^{×}. (Hint: try a few examples.)

2. Describe a 2-Sylow subgroup of S_{8}. How many are there?

3. Describe all 2-Sylow subgroups of S_{4}. Identify this group.

4. Do one problem from Problem Set 3\frac{1}{2}.

5. (Dummit and Foote p. 169) For any group G define the dual group of G (denoted Ĝ) to be the set of all homomorphisms from G into the multiplicative group of roots of unity in ℂ. Define a group operation in Ĝ by pointwise multiplication of functions: if χ, ψ are homomorphisms from G into the group of roots of unity then χψ is the homomorphism given by (χψ)(g) = χ(g)ψ(g) for all g ∈ G, where the latter multiplication takes place in ℂ.

(a) Show that this operation on Ĝ makes Ĝ into an abelian group. [Show that the identity is the map g ↦ 1 for all g ∈ G and the inverse of χ ∈ Ĝ is the map g ↦ χ(g)^{−1}.]

(b) If G is a finite abelian group, prove that Ĝ ≅ G. [Write G as ⟨x_{1}⟩ × ⋯ × ⟨x_{r}⟩ and if n_{i} is the order of x_{i} define χ_{i} to be the homomorphism which sends x_{i} to e^{2πi/n_{i}} and sends x_{j} to 1, for all j ≠ i. Prove χ_{i} has order n_{i} in Ĝ and Ĝ = ⟨χ_{1}⟩ × ⋯ × ⟨χ_{r}⟩.]

This result is often phrased: a finite abelian group is self-dual. It implies that the lattice diagram (of subgroups) of a finite abelian group is the same when it is turned upside down. Note however that there is no natural isomorphism between G and its dual; the isomorphism depends on a choice of a set of generators for G. This is frequently stated in the form: a finite abelian group is noncanonically isomorphic to its dual.

The set is due Tuesday, November 12 at 3:30 pm in Pierre Albin’s mailbox (opposite the elevator on the first floor of Building 380).

Date: Tuesday, November 5, 2002.