Groups, Rings and Fields: Problem sheet 2 Solutions must be submitted by Monday 4 November 2002

1. Give the operation tables of the cyclic groups of orders 6, 7 and 8. In each case find as many subgroups as possible. [14 marks]

2. Recall that the set of real numbers R does not form a group under multiplication. However if we discard 0 and consider the set R^{∗} of nonzero real numbers we do get a multiplicative group.

Similarly, the set Z_{m} = {0, 1, 2 …, m − 1} doesn’t form a group under multiplication modulo m. For the cases m = 5, m = 6 and m = 8 investigate whether you can discard elements from Z_{m} to leave a multiplicative group. [12 marks]

3. Use your answer to question 1 to predict the number of subgroups of (a) the cyclic group of order 24, (b) the cyclic group of order 60, (c) the cyclic group of order 84. [8 marks]

4. Recall that the dihedral group D_{n} has the identity I and is generated by elements R and S satisfying R^{n} = I, S^{2} = I and SR = R^{−1}S.

Find all the left cosets and the right cosets of the following subgroups of D_{6}:

(a) H_{1} = {I, R^{3}},

(b) H_{2} = {I, R^{2}, R^{4}},

(c) H_{3} = {I, R^{3}, S, R^{3}S}. [12 marks] 5 In each case calculate στ, τσ, σ^{2}, τ^{−1} and τ^{−1}στ. Also find the order and sign of each permutation you calculate:

(a) σ = (1 2 3 4 5 6 7 8 9 10), τ = (1 10)(2 9)(3 8)(4 7)(5 6);

(b) σ = (2 6 9 4)(3 10 7 5), τ = (1 4 7)(2 5 8)(3 6 9);

(c) σ = (1 10 2 9 3 8 4 7 5 6), τ = (3 7 9 5)(4 6 10 8).

Do a few more examples of your own choosing, until you are completely confident about calculating with permutations.

Note that στ ≠ τσ in general, but do στ and τσ have anything in common?

6. Write down the multiplication table of S_{3}. [10 marks]

7. Write down all (twenty-four) elements of S_{4} in cycle notation. [10 marks]

8. Find all the left cosets and the right cosets of the following subgroups of S_{4}:

(a) H_{1} = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)},

(b) H_{2} = {id, (1 2 3 4), (1 3)(2 4), (1 4 3 2)},

(c) H_{3} = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3), (1 2), (3 4), (1 3 2 4), (1 4 2 3)}. [18 marks] 9. A magician takes a pack of twenty cards and performs a complete riffle shuffle on them (as in the 8-card example in the lecture). If the cards start in order

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20  (∗)

they finish in order

11 1 12 2 13 3 14 4 15 5 16 6 17 7 18 8 19 9 20 10.

Write down, in cycle notation, the permutation corresponding to this shuffle and hence determine how many times it needs to be repeated to bring the cards back to their original order.

The magician now employs a different shuffle; after performing the riffle shuffle, he moves the top five cards to the bottom of the pack. Thus starting in the order (∗) the cards finish in order

3 14 4 15 5 16 6 17 7 18 8 19 9 20 10 11 1 122 13.

How many times must this new shuffle be repeated to bring the pack back to its original order? [16 marks]

10. In MAPLE there is a group theory package which includes commands for multiplying and inverting permutations. To use first load the package using

> with(group):

To enter a permutation σ = (1 5 3)(2 6) we write

> sigma:=[[1, 5, 3], [2, 6]]:

The inverse of σ is obtained using the invperms command:

> invperms(sigma);
        [[1, 3, 5], [2, 6]]

To multiply two permutations, say σ and τ = (1 4)(2 5)(3 6) we use the mulperms command. To obtain στ we must enter mulperms(tau, sigma) (note the change of order!)

> tau:=[[1, 4], [2, 5], [3, 6]]:
> mulperms(tau, sigma);
        [[1, 4, 5, 6], [2, 3]]

Let σ = (1 2 3 4 5 6 7 8 9 10 11) and τ = (1 12)(2 7)(3 9)(4 5)(6 8)(10 11). Use MAPLE to calculate σ^{3}τσ^{2}τ and σ^{−2}τσ^{−1}τσ^{3}τ.

RJC 17/10/2002