Masters Algebra Examination Oklahoma State University Department of Mathematics January, 1998

1. Let G be a finite group and x and y elements of G. Let a be the order of x and b be the order of y. It is given that a and b are relatively prime.

(a) Prove: if G is abelian, then the subgroup generated by x and y has order ab.

(b) Prove: if G is abelian, then the subgroup generated by xy has order ab.

(c) By way of example, show that (a) and (b) do not necessarily hold in a non-abelian group.

2. Consider S_{n}, the symmetric group on n objects.

(a) Show that any two r–cycles in S_{n} are conjugate.

(b) If (a_{1} a_{2} ⋯ a_{r}) is an r–cycle in S_{n}, what is the size of its conjugacy class?

(c) Determine the center of S_{n}.

3. Let V be a vector space over a field F, and let T : V → V be a linear transformation of V into itself. Suppose there are non-zero vectors v_{1}, …, v_{m} and distinct scalars c_{1}, …, c_{m} (i.e., c_{i} ≠ c_{j} for i ≠ j) such that T (v_{j}) = c_{j}v_{j} for each j = 1, …, m. Prove that the vectors v_{1}, …, v_{m} are linearly independent.

4. Let R be an integral domain together with a function σ : R → {0, 1, 2, 3, ...} satisfying:

(a) σ(a) = 0 if and only if a = 0;

(b) For any a,b ∈ R, a ≠ 0, either b = qa for some q ∈ R or there are x,y ∈ R such that 0 < σ(ax + by) < σ(a).

Prove R is a principal ideal domain.

5. (a) Let K ⊂ F ⊂ E be a tower of field extensions of finite degree. Prove that [E : K] = [E : F] [F : K].

(b) Let E/F be a field extension of finite degree, and suppose E = F (α) for some α ∈ E. If the degree [E : F] is odd, prove that E = F (α^{2}).

6. Let Λ be the ℤ-submodule of ℤ^{3} generated by (2, 3, −1) and (1, 0, −1). Express ℤ^{3}/Λ as a direct product of cyclic groups.