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

1. The following statements can be easily proved simply from knowledge of the definitions of the terms subgroup, homomorphism, kernel and image. Do so. Although some have appeared in the lectures, you should not look at the proofs I gave, but should construct your own.

(i) The kernel of a group homomorphism ϕ : G_{1} → G_{2} is a subgroup of G_{1}. (ii) The image of a group homomorphism ϕ : G_{1} → G_{2} is a subgroup of G_{2}. (iii) Let H be a subgroup of G and K be a subgroup of H. Show that K is a subgroup of G. [9 marks]

2. Let G = S_{4} and H = {id, (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)}. In question 6 of sheet 2 you calculated the left and right cosets of H in G and found they were the same. In other words, H is a normal subgroup of G. Find the multiplication table of G/H. Is G/H isomorphic to a group you are already familiar with? [8 marks]

3. This example shows that normality of the subgroup is necessary in forming quotient groups. Let G = S_{3} and H = {id, (1 2)}. Write down the three left cosets of H in G. Call them H, A and B. Find σ_{1}, σ_{2} ∈ A with σ_{1}σ_{2} ∈ B and find τ_{1}, τ_{2} ∈ A with τ_{1}τ_{2} ∈ H. [6 marks]

4. Let p denote an odd prime number, and suppose there is an integer a with

a^{2} ≡ −1 (mod p).

Show that the cyclic subgroup H = ⟨a⟩ of Z_{p}^{∗} has order 4, and hence prove that p ≡ 1 (mod 4). [6 marks]

5. Consider the multiplicative group Z_{11}^{∗}. Then we calculate 2^{1} ≡ 2, 2^{2} ≡ 4, 2^{3} ≡ 8, 2^{4} ≡ 5, 2^{5} ≡ 10, 2^{6} ≡ 9, 2^{7} ≡ 7, 2^{8} ≡ 3, 2^{9} ≡ 6, and 2^{10} ≡ 1 (mod 11). Hence Z_{11}^{∗} = ⟨2⟩.

In each case find a such that Z_{p}^{∗} = ⟨a⟩: (i) p = 7, (ii) p = 13, (iii) p = 17. [9 marks]

6. Using the ring axioms one can prove that familiar identities such as 0a = 0 are in fact true in every ring. For instance we prove this as follows:

0 + 0 = 0  [A3]
(0 + 0)a = 0a
0a + 0a = 0a  [D]
(0a + 0a) + (−(0a)) = 0a + (−(0a))
0a + (0a + (−(0a))) = 0a + (−(0a)) [A2]

0a + 0 = 0  [A4]
0a = 0  [A3].

If you like this sort of thing, then prove that a0 = 0, −a = (−1)a, −a = a(−1), −(ab) = (−a)b and −(ab) = a(−b) are valid for all elements a, b in any ring R.

7. Suppose that in the ring R we have 1 = 0. Using the result proved in question 6 prove that R has exactly one element. [2 marks]

8. In each case determine whether or not S is a subring of R. When S is a subring, determine whether it is commutative. When S is a commutative subring, determine whether or not it is a field.

(i) R = Q, S = {m/n : m, n ∈ Z, n odd}. (ii) R = R, S = \{a + b\sqrt{3} : a, b ∈ Z\}. (iii) R = R, S = \{a + b ^{3} \sqrt{2} : a, b ∈ Z\}. (iv) R = C(R), the ring of continuous functions from R to R, S = {f ∈ C(R) : f is differentiable}.

(v) R = M_{2}(R),

S = \{( \begin{matrix}a & 0 \\ c & d \end{matrix} ) : a, c, d ∈ R\} .

(vi) R = M_{2}(R),

S = \{( \begin{matrix}a & −b \\ b & a \end{matrix} ) : a, b ∈ R\} .

(vii) R = M_{2}(Q),

S = \{( \begin{matrix}a & 2b \\ b & a \end{matrix} ) : a, b ∈ Q\} .

(viii) R = M_{2}(R),

S = \{( \begin{matrix}a & 2b \\ b & a \end{matrix} ) : a, b ∈ R\} .

(ix) R = R[X], S = {a_{0} + a_{1}X + ⋯ + a_{n}X^{n} : n ∈ N, a_{0} = 0}.

(x) R = R[X], S = {a_{0} + a_{1}X + ⋯ + a_{n}X^{n} : n ∈ N, a_{1} = 0}. [50 marks]

9. Consider the matrices

M_{1} = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , M_{i} = ( \begin{matrix}i & 0 \\ 0 & −i \end{matrix} ) , M_{j} = ( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} ) , M_{k} = ( \begin{matrix}0 & i \\ i & 0 \end{matrix} ) .

Show that these obey the rules corresponding to quaternion multiplication. That is, the rule corresponding to i^{2} = −1 is M_{i}^{2} = −M_{1}, to ij = k is M_{i}M_{j} = M_{k} etc. Deduce that quaternion multiplication is associative. [10 marks]

RJC 18/11/2002