Groups, Rings and Fields: Problem sheet 1 Solutions must be submitted by Monday 21 October 2002

0. Visit the course website at

www.maths.ex.ac.uk/∼rjc/courses/grf02/grf02.html

and bookmark it.

1. Draw an equilateral triangle. Describe each of its symmetry operations. Complete the multiplication table of its symmetry group. Find as many of its subgroups as possible. [12 marks]

2. Draw a regular hexagon. Describe each of its symmetry operations. Complete the multiplication table of its symmetry group. Find as many of its subgroups as possible. [20 marks]

3. Which of these sets form a group under the given operation? Give reasons.

(a) N = {1, 2, 3, …} with the operation of addition.

(b) G_{1} = {0, 1, 2, 3, …} with the operation of addition.

(c) Z with the operation a ∗ b = a + b − 1.

(d) G_{2} = {a ∈ R : a ≠ 0} with the operation a ∗ b = 3ab.

(e) G_{3} = {a ∈ R : a ≠ 1} with the operation a ∗ b = a + b − ab. [20 marks]

4. In each case I give a group G and a subset H of G. Determine whether or not H is a subgroup of G. Give reasons.

(a) G = R^{∗} with the operation of multiplication, H = {x ∈ R : x ≥ 1}.

(b) G = Sq with the operation of composition, H = {S_{a}, S_{b}, S_{c}, S_{d}}, the set of reflections in Sq.

(c) G = C^{∗} with the operation of multiplication, H = {z ∈ C : |z| = 1}. [10 marks]

5. In each group the identity element e satisfies e ∗ e = e. Is there a group G and a non-identity element x ∈ G with x ∗ x = x? Give reasons. [6 marks]

6. Can a group have more than one identity element? Give reasons. [6 marks]

7. Can an element a of a group G have more than one inverse? Give reasons. [6 marks]

8. Consider a cube. (It may help in this question to obtain a cube of some description). How many rotational symmetry operations has it? What are the orders of these rotations, and how many rotations have each possible order? [Hint: consider the possible axes of rotation.] [20 marks]

RJC 4/10/2002