189-235A: Basic Algebra I Assignment 5

Due: Wednesday, November 28

1. Let f : Z ⟶ R be a surjective homomorphism of rings. Show that R is isomorphic either to Z or to the ring Z/nZ for a suitable n ≥ 1.

2. Let R be a commutative ring and let I be an ideal of R. Prove or disprove the statement that if R is an integral domain, then so is R/I.

3. 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.

4. 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 in an infinitesimal neighbourhood of the value x = 0”.)

5. 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.)

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

7. 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. (Hint: try to find such a group in the group of 3 × 3 invertible matrices with entries in Z/3Z.)

8. 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?

9. 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/pZ)^{×} (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.

10. 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.

11. 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, by writing down an explicit isomorphism between the two groups.

12. Recall that 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}.