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SIM2004/SIM2014 ALGEBRA I ASSIGNMENT, SEMESTER 1, 2023/2024 ACADEMIC SESSION
发布时间：2023-12-19
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SIM2004/SIM2014 ALGEBRA I
ASSIGNMENT,
SEMESTER 1, 2023/2024 ACADEMIC SESSION
SECTION A: Answer ALL questions.
1. Determine whether the following are true or false. If it is false, justify your answer.
(a) Every subgroup of an abelian group is normal.
(b) If G is a group of order 2024 with subgroup H of order 23, then the number of right cosets of H in G is 44.
(c) Let G be a group and N G. Then the group G/N always exists.
(d) Let (G, ·) and (H, +) be two groups and ϕ : G → H be an onto group homo- morphism. If suppose ker ϕ = {1}, then G = H.
2. Given that (Z, *) is group with respect to the following binary operation
x * y = 2 + x + y for all x, y ∈ Z.
Determine whether (Z, *) is a group. If not, explain why. If it is a group, determine whether it is an abelian group.
(Hint: Note that Z is indeed closed with respect to * and “+” is the usual addition in Z.)
3. Consider the cyclic group C30 = hx | x30 = 1i.
(a) What are the possible orders of subgroups of C30 ?
(b) Find two subgroups H and K of C30 such that H ∩ K = {1}. (c) Find a subgroup L of C30 such that [C30 : L] = 5.
4. Let G be a group with H, K G such that H G and H ∩ K K. Given that [K : H ∩ K] = m[H : H ∩ K]. Find the value of λ in terms of m, so that [G : K] = λ[G : H].
(Hint: [G : K] = |G/K| = |G|/|K| .)
5. Given G is a inite group, H G, and K G such that K < H.
(a) Show that H/K G/K.
(Hint: Use either Criterion S1 or Criterion S2.)
(b) If |G/K| = p where p is a prime number, show that |H| = p|K| . (Hint: Use Lagrange’s Theorem.)
6. Given G = D8 = hx, y | x4 = y2 = 1, yx = x3yi is a non-abelian group with the following Cayley Table.
Let H = {1, x, x2 , x3 } and L = {1, x2 }. By referring to the above table, answer the following questions.
(a) Is H G? Justify your answer.
(b) Find [G : H]. Hence, by listing all the right and left cosets of H in G, determine whether H G.
(c) Find [G : L]. Hence, by listing all the right and left cosets of L in G, show that L G.
(d) Find the elements of G/L.
7. Given an additive group of integers, Z. Consider the quotient group Z/pZ where p is a prime number. The element x ∈ Z/pZ can be written as x = z + pZ for some z ∈ Z. Let θ : Z → Z/pZ be deined as
θ(z) = z + pZ for all z ∈ Z.
(Caution! Beware of the additive binary operation on homomorphism and cosets.)
(a) If θ is well-deined, show that θ is an onto homomorphism.
(b) Is θ an isomorphism? Justify your answer.
(c) Find ker θ .
