Math 551, Assignment 6, due Wednesday, November 20 in class

1. Suppose that the square matrix A is in block diagonal form:

A = [ \begin{matrix}A_{1} & 0 & ⋯ & 0 \\ 0 & A_{2} & ⋯ & 0 \\ & & ⋯ \\ 0 & 0 & ⋯ & A_{r} \end{matrix} ] .

Show that ν(A) = ∑_{i=1}^{r} ν(A_{i}), μ_{A} = lcm_{i=1}^{r} μ_{A_{i}} and χ_{A} = ∏_{i=1}^{r} χ_{A_{i}}. Here ν(A) is the nullity (dimension of the nullspace) of A, and μ_{A} and χ_{A} are the minimal and characteristic polynomials, respectively. Which of these statements remain true if nonzero entries are allowed anywhere above the diagonal?

2. Let A ∈ Mat_{6×6}(C) and suppose that (A + I)^{12} = 0. Show that (A + I)^{6} = 0. Give a set of representatives for the similarity classes of all such matrices A. Explain briefly how the answer would differ (if at all) over an arbitrary field k instead of C.

3. Let A be a square matrix over a field k. Show that μ_{A} = μ_{A^{T}} (A^{T} being the transpose of A). Show that A is similar to A^{T}. (Hint. First do the case A = C(f) for some monic polynomial f.)

4. Let H ≤ G, with G and H both free abelian groups of the same finite rank. Let A be the matrix of the inclusion mapping α : H → G with respect to some bases B and B′ of G and H respectively (i.e. the columns of A are the B-coefficients of the elements of H). Show that |G/H| = ± det(A).

5. Let G be a finitely generated free abelian group, and let H ≤ G. Show that there is a unique subgroup H^{∗} ≤ G such that (a) H ≤ H^{∗}; (b) rank(H) = rank(H^{∗}); and (c) G = H^{∗} ⊕ K for some subgroup K.

6. An elementary matrix is a square matrix which is the same as the identity matrix except possibly in a single off-diagonal entry. Let R be a Euclidean domain. Show that a square matrix A ∈ R^{m×m} satisfies det(A) = 1 if and only if A is the product of elementary matrices in R^{m×m}. Deduce that A is invertible in R^{m×m} if and only if A is the product of elementary matrices and a diagonal matrix whose entries are units in A.

7. Isomorphism-invariant data obtained from all objects in a class (e.g. all abelian groups) are called invariants of those objects. A set of invariants is called complete when it is true that two objects are isomorphic if and only if they have the same invariants. Which of the following sets of invariants of a finite abelian group G is complete? Give proof or counterexample.

a) {|G|, the exponent of G, the rank of G} (rank=smallest number of direct summands possible in an expression for G as the direct sum of cyclic groups).

b) {n_{i} | i ∈ Z^{+}}, where for any positive integer i, n_{i} is the number of elements of G of order i.

8. Let G be a finite subgroup of GL_{n}(Q), where Q is the field of rational numbers. Show that G is conjugate (in GL_{n}(Q)) to a subgroup of GL_{n}(Z). (Hint. Take any Q-basis B of the Q-vector space Q^{n} of n × 1 column vectors, and consider the abelian group generated by all gb, g ∈ G, b ∈ B.)