189-235A: Basic Algebra I Assignment 8

Due: Wednesday, November 23

1. 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.

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

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

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

5. 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.

6. Show that the groups S_{3} and GL_{2}(Z_{2}) are isomorphic, by writing down an isomorphism between them.

7. A transposition in the symmetric group S_{n} is a permutation of the form (ab) (i.e., a permutation that interchanges two elements a, b ∈ {1, …, n}, leaving all other n − 2 elements fixed. Show that every permutation in S_{n} can be expressed as a product of transpositions.