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MATH2022: Linear and Abstract Algebra Semester 1, 2026
Assignment 1
This individual assignment is due by 11:59pm Friday 27 March 2026, via Canvas. Late assignments will receive a penalty of 5% per day until the closing date. A single PDF copy of your answers must be uploaded in the Assignments tab on Canvas https://canvas.sydney.edu.au/courses/71507/assignments/679878. Please make sure you review your submission carefully. It is your responsibility to ensure that your uploads are successful and that the uploaded files are clear, legible, and correctly oriented. Hart to read, illegible, or sideways/upside down submissions might not be marked. What you see is exactly how the marker will see your assignment. Submissions can be overwritten until the due date.
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1. Let G be a group and H ≤ G a subgroup of G. For any element g ∈ G, the coset gH is defined as the collection of products gH = {gh| h ∈ H}.
Many elements g, g′ ∈ G with g = g
′
can satisfy an equality of cosets gH = g
′H.
(a) Let G = S3 and H = A3. List all possible distinct cosets of the form σA3 with σ ∈ S3 and their elements.
(b) We say a subgroup H ≤ G is a normal subgroup if for all g ∈ G and all h ∈ H, ghg−1 ∈ H.
(i) Justify why An is a normal subgroup of Sn for any n.
(ii) Let G be an abelian group. Justify why any subgroup H ≤ G is a normal subgroup.
(c) Let H ≤ G be a normal subgroup. We define a group operation on the set of cosets
{gH} as follows: for any two cosets g1H and g2H,
(g1H)(g2H) = (g1g2)H.
To see that this operation is “well-defined,” prove just the following necessary fact: for any choice of elements g1h ∈ g1H and g2h
′ ∈ g2H, the product (g1h)(g2h
′
) is in the coset (g1g2)H.
Remark: We denote this group of cosets {gH} by G/H, and call it the quotient group of G by H. The other group axioms for G/H follow quickly from the group axioms for G, so we won’t verify them on this assignment.
(d) Compute the multiplication table for the quotient group S3/A3 based on your answer in part (a). What cyclic group is S3/A3 isomorphic to?
2. Another common notation for permutations σ ∈ Sn that isn’t disjoint cycles is known as one-line notation. This keeps track of where the permutation σ sends each of 1, 2, 3, . . . , n in order. For example, in S3 the permutation σ = (1 2 3) has one-line notation 231, since σ sends 1 7→ 2, 2 7→ 3, 3 7→ 1. In S5, the permutation σ = (2 4 3)(1 5) has one-line notation 54231, since 1 7→ 5, 2 7→ 4, 3 7→ 2, 4 7→ 3, 5 7→ 1. In any Sn, the identity permutation e has one-line notation 1234 · · · n.
(a) Translate the following permutations in S5 from one-line notation to cycle notation.
(i) 35412
(ii) 34125
(iii) 54321
(iv) 14532
(b) For a permutation σ in one-line notation, say σ = x1x2 . . . xn, an inversion is a pair of indices i < j with xi > xj
. We write this as [i, j]. For example, the permutation 231 has two inversions: [1, 3], since 1 < 3 and x1 = 2 > x3 = 1, and [2, 3], since 2 < 3 and x2 = 3 > x3 = 1. But, [1, 2] is not an inversion since 1 < 2
and x1 = 2 < x2 = 3.
Find the inversions of each permutation in S5 from part (a).
(c) Another important piece of data for a permutation σ is it’s length: the length of σ, written ℓ(σ), is the number of inversions of σ. For example, we saw in S3 the one-line permutation σ = 231 had two inversions, so we write ℓ(σ) = 2.
(i) What is the length of each of the permutations in part (a)?
(ii) What pattern do you see between the length of a permutation and its parity?