189-235A: Basic Algebra I Assignment 8

Due: Wednesday, November 10

1. Let F be a field, and let f : F ⟶ R be a homomorphism satisfying f(0) ≠ f(1). Show that f is necessarily injective, and that the image of f is isomorphic to F.

2. Show that the image of Z under any homomorphism is isomorphic either to Z or to a finite ring.

Let R be a commutative ring and let I be an ideal of R. Prove or disprove the following statements.

3. If R is an integral domain, then so is R/I.

4. If every ideal in R is principal, the same is true of R/I.

5. If every ideal in R/I is principal, the same is true of R.

6. Let R = Z[x], and let I be the ideal (p, x^{2} + 1) generated by the integer prime p and the polynomial x^{2} + 1. Show that R/I is isomorphic to Z/5Z × Z/5Z is p = 5, and is isomorphic to a field with 49 elements if p = 7.

Extra credit.

7. Let F be a field and let R = F [[x]] denote the ring of formal power series with coefficients in F, i.e., the set of expressions of the form

∑_{n=0}^{∞} a_{n}x^{n}, a_{n} ∈ F,

where the addition and multiplication are performed by formally expanding out the sums and products (without worrying about issues of convergence, which don’t make sense in an arbitrary field F anyways!)

Let I = (x) be the ideal generated by the power series x. Show that R/I is isomorphic to F. Show that any element of R which does not belong to I is invertible. Conclude that any non-trivial ideal of R is contained in I. (A ring with this property is called a local ring, a terminology arising from the prototypical example F [[x]], because power series can be thought of as “functions defined only around the value x = 0”.)

1. Show that the intersection of two prime ideals need not be a prime ideal.

2. Show that a ring R is an integral domain if and only if {0} is a prime ideal.

3. If P is a prime ideal in a commutative ring R, is the ideal P × P a prime ideal of R × R?

4. Find an ideal of Z × Z that is prime but not maximal.

5. Let R be a commutative ring, let F [x] be the ring of polynomials with coefficients in a field F, and let f : R ⟶ F [x] be a ring homomorphism. Show that the kernel of f is a prime ideal of R.

6. Let F be a field, and define a binary composition law on G = F − {1} by the rule

a ∗ b = a + b − ab.

Show that G, with this operation, is a group. (In particular, write down the neutral element for ∗, and give a formula for the inverse of a ∈ G.

7. List all the elements of order 3 in S_{3}. How many are there?

8. List all the elements of order 6 in S_{5}. How many such elements are there?

9. Given an example of non-abelian groups of order 12 and 30.

10. Extra credit question. Suppose that G is a group in which x^{2} = 1, for all x ∈ G. Show that G is abelian. Give an example of a non-abelian group G of order 27 in which x^{3} = 1 for all x ∈ G.

1. Show that the conjugate elements a and bab^{−1} in a group G have the same order.

2. Show that the intersection of two subgroups H_{1} and H_{2} of a group G is a subgroup of G. What about unions of subgroups?

3. If a is an element of a finite group G of cardinality n, show that a^{n} = 1. Apply this general fact to the group G = Z_{p}^{×} (under multiplication) to give another proof of Fermat’s Little Theorem that p divides a^{p} −a for all integers a when p is prime.

4. Let S be a subset of a group G. The centraliser of S, denoted Z(S), is the set of a ∈ G which commute with every s ∈ S, i.e., such that as = sa for all s ∈ S. Show that Z(S) is a subgroup of G.

5. Let G_{1} be the group of strictly positive real numbers, under multiplication, and let G_{2} be the group of all real numbers, under addition. Show that G_{1} and G_{2} are isomorphic.

6. Let f : G_{1} ⟶ G_{2} be a homomorphism of groups, and let a be an element of G_{1} of finite order. Show that the order of f(a) in G_{2} divides the order of a in G_{1}, and that these two orders are equal if f is injective.

7. Recall the ring H = {a + bi + cj + dk, a, b, c, d ∈ R} of quaternions defined by the multiplication rules

i^{2} = j^{2} = k^{2} = −1, ij = −ji = k, ki = −ik = j, jk = −kj = i.

Show that the subset G = {1, −1, i, −i, j, −j, k, −k} is a subgroup of the multiplicative group of non-zero elements of H. Show that G is not isomorphic to the dihedral group D_{8} of order 8.

Extra Credit

9. Show that the groups GL_{2}(Z_{2}) and S_{3} are isomorphic, by writing down a specific isomorphism. (Hint: realize each matrix as a permutation on the non-zero column vectors with entries in Z_{2}).

10. The conjugacy class of a in a group G is the set of all elements of G which are of the form gag^{−1} for some g ∈ G. Show that a normal subgroup of G is a disjoint union of conjugacy classes. List the conjugacy classes in S_{4} and use this to give a complete list of all the normal subgroups of S_{4}. Same question for S_{5}.