Math 551, Assignment 6

Due Thursday, November 9, in class

1. Fix a prime p. Let G = ∏_{n=1}^{∞} Z_{p^{n}}, and let H = ⊕_{n=1}^{∞}Z_{p^{n}} ≤ G. Show that for every g ∈ T (G) there exists g_{1} ∈ T (G) such that g − pg_{1} ∈ H. Conclude that T (G)/H is not the direct sum of cyclic groups.

2. 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.

3. 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.

4. For any m × n matrix A (square or not), and any positive integer r ≤ min(m, n), an r × r minor of A is defined as an r × r matrix obtained by deleting some rows and columns of A. If A has all its entries in a Euclidean domain R, define Δ_{r}(A) to be the g.c.d. of the determinants of all its r × r minors (this is well-defined up to association). Use the previous exercise and the fundamental theorem to show that the invariant factors of A are

Δ_{1}(A), Δ_{2}(A)/Δ_{1}(A), ⋯, Δ_{r}(A)/Δ_{r−1}(A),

where r is the greatest integer such that Δ_{r}(A) ≠ 0, i.e., r is the rank of the matrix A. [A consequence is that for two m × n matrices over the Euclidean domain R, A ∼ B ⟺ Δ_{r}(A) and Δ_{r}(B) are associates for each 1 ≤ r ≤ min(m, n).]

5. Let H ≤ G, with G and H both free abelian groups of the same 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).

6. 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.

7. Let M be a finitely generated module over a PID R, and let N be a submodule of M. Relate the invariant factors of N to those of M.

8. Show that the following hold for arbitrary finitely generated modules M, N, P over a PID: (a) M ⊕ P ≅ N ⊕ P implies M ≅ N; (b) If M is isomorphic to a submodule of N, and N is isomorphic to a submodule of M, then M ≅ N. Give counterexamples to show that the assumption of finite generation is necessary.

9. Let M be an abelian group. Show that the abelian group Hom(M, M), together with the operation of composition of functions, is a ring (with 1). Show that an R-module is an abelian group M together with a ring homomorphism ϕ : R → Hom(M, M), i.e., an R-module determines such a pair (M, ϕ), and such a pair determines an R-module. Then give a 1-line proof that if M is an R-module and A is an ideal such that am = 0 for all a ∈ A and m ∈ M, then M is an R/A-module.

10. 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.)