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

1. Let S_{n} denote the symmetric group on n objects.

(a) Give exactly one representative of each conjugacy class of S_{6}.

(b) Consider the element σ = (1 2 3)(4 5 6) of S_{6} (written in its disjoint cycle decomposition). Determine:

i. the order of σ;

ii. the order of the centralizer Z(σ) of σ in S_{6};

iii. the size of the conjugacy class of σ; iv. the structure of the subgroup Z(σ) of S_{6}.

2. Consider groups of order 1998 = 2 × 3^{3} × 37.

(a) Determine all Abelian groups of order 1998.

(b) Can a group of order 1998 be simple? Prove your answer.

(c) Can a group of order 1998 be non-Abelian? Prove your answer.

3. Let V be a vector space of dimension n over a field F. Let

0 = W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{n−1} ⊂ W_{n} = V

be a sequence of subspaces such that dim(W_{i}) = i for 0 ≤ i ≤ n. Suppose T : V → V is a linear operator for which each W_{i} is an invariant subspace. Prove there is a basis B of V such that the matrix of T with respect to B is upper triangular.

4. (a) State the definitions of Principal Ideal Domain and Unique Factorization Domain.

(b) Give examples of each of the following:

i. An integral domain which is not a unique factorization domain. Justify. ii. A unique factorization domain which is not a principal ideal domain. Justify.

5. For each of the following groups G, give a field extension K of ℚ that is Galois over ℚ, with Galois group G_{K/ℚ} of K over ℚ isomorphic to G. Prove your claims.

G = ℤ/2ℤ , ℤ/2ℤ × ℤ/2ℤ , S_{3} , D_{4}