Groups, Rings and Fields: Problem sheet 3 Solutions must be submitted by Monday 18 November 2002

1. Express the following permutations as products of transpositions:

(i) (1 4 7 2 5 8 3 6); (ii) (1 10 7 4)(2 3 9 8 5); (iii) (1 12 4 8 7 3 6)(2 11 10 5). [6 marks]

2. Show that if the permutation σ has odd order m, then σ is an even permutation. (Hint: calculate sgn(σ^{m}) in two ways.) [6 marks]

3. Let H_{1} and H_{2} be subgroups of the group G. Show that H_{1} ∩ H_{2} is also a subgroup of G.

Find subgroups H_{1} and H_{2} of S_{3} such that H_{1} ∪ H_{2} is not a subgroup of S_{3}. [6 marks]

4. Let G be a multiplicative abelian group. Show that ϕ : G → G and ψ : G → G defined by ϕ(a) = a^{−1} and ψ(a) = a^{2} are homomorphisms.

If, however, G is the non-abelian group S_{3}, show that ϕ and ψ are no longer homomorphisms. (Hint: to show that ϕ is not a homomorphism it suffices to find σ and ρ in S_{3} with ϕ(σρ) ≠ ϕ(σ)ϕ(ρ).) [10 marks]

5. Write down the multiplication table of the group A_{4}. (This has 12 elements: the even permutations in S_{4}). Find subgroups of A_{4} of orders 1, 2, 3, 4 and 12. Can you find a subgroup of order 6? [15 marks]

6. Let G_{1} and G_{2} be (multiplicative) groups with identities e_{1} and e_{2} respectively. Let ϕ : G_{1} → G_{2} be a homomorphism. Show that:

(i) ϕ(e_{1}) = e_{2}; (ii) ϕ(a^{−1}) = ϕ(a)^{−1} for each a ∈ G_{1}. [Hint: it may help to recall questions 6 and 7 of sheet 1.] [10 marks]

7. In each case, given groups G_{1} and G_{2} and a map ϕ : G_{1} → G_{2} determine, giving reasons, whether the map ϕ is a homomorphism. In each case, if ϕ is a homomorphism, describe its kernel and image in as much detail as possible:

(i) G_{1} = C^{∗}, G_{2} = C^{∗} (both multiplicative groups), ϕ(z) = z/|z|; (ii) G_{1} = Z_{4} = {0, 1, 2, 3} (under the operation of addition modulo 4), G_{2} = Z_{2} = {0, 1} (under the operation of addition modulo 2), ϕ defined by

ϕ(0) = 0, ϕ(1) = 1, ϕ(2) = 0, ϕ(3) = 1

(so that ϕ(a) is the remainder left after dividing a by 2); (iii) G_{1} = Z_{5} = {0, 1, 2, 3, 4} (under the operation of addition modulo 4), G_{2} = Z_{2} = {0, 1} (under the operation of addition modulo 2), ϕ defined by

ϕ(0) = 0, ϕ(1) = 1, ϕ(2) = 0, ϕ(3) = 1, ϕ(4) = 0

(so that ϕ(a) is the remainder left after dividing a by 2); (iv) G_{1} = R (under the operation of addition), G_{2} = GL_{2}(R) (under the operation of multiplication),

ϕ(a) = ( \begin{matrix}1 + a & a \\ −a & 1 − a \end{matrix} );

(v) G_{1} = R (under the operation of addition), G_{2} = GL_{3}(R) (the group of 3 by 3 nonsingular matrices under the operation of multiplication),

ϕ(a) = ( \begin{matrix}1 & a & a^{2} \\ 0 & 1 & 2a \\ 0 & 0 & 1 \end{matrix} ) .

(vi) G_{1} = GL_{2}(R) G_{2} = GL_{2}(R) (both under the operation of multiplication), ϕ(A) = A^{t} (the transpose of A). (vii) G_{1} = R (under the operation of addition), G_{2} = GL_{2}(R) (under the operation of multiplication),

ϕ(t) = ( \begin{matrix}cos t & − sin t \\ sin t & cos t \end{matrix} );

(viii) G_{1} = C^{∗} (under the operation of multiplication), G_{2} = GL_{2}(R) (under the operation of multiplication),

ϕ(a + ib) = ( \begin{matrix}a & b \\ −b & a \end{matrix} ) .

[47 marks]

RJC 1/11/2002