Assignment 1: Permutations, Subgroups, and Isometries

Question 1: Permutations

Deadline for submission is 4pm Friday October 29th, 2021. Assignments should be submitted via MyAberdeen.

Justify your answers.

1. (a) Let α: {1, . . . , 9} → {1, . . . , 9} be the permutation defifined by

Write it as a product of disjoint cycles. [4 marks]

(b) Let τ = (4 11 2 13)(1 7 9 3 10 15)(8 5 6 12) be a permutation in S15. Compute τ 16 . [4 marks]

(c) Find a permutation α such that α ? (1 7 6 8 3) = τ . [4 marks]

(d) Find all the permutations a in S4 such that a = a−1. [4 marks]

2 (a) Show that if H and K are subgroups of G  then H ∩ K is a subgroup of G, and hence also a subgroup of H and of K. [4 marks]

(b) Let G  be a group. Show that if H, K  ≤ G  are subgroups of order 3 then either H = K  or H ∩ K = {1} (the trivial subgroup). [4 marks]

3. Recall that Iso(R2 ) denotes the group of isometries of the plane R2.

(a) Let A be a subset of R2 and set

Iso(R2 , A) = ? ∈ Iso(R2 ) : ?(A) = A .

Prove that Iso(R2 , A) is a subgroup of Iso(R2 ). [4 marks]

(b) By defifinition, if ? ∈ Iso(R2 , A) then the restriction of ? to A gives a well defifined function ?|A : A → A. Prove that ?|A is bijective, namely it is an element of Sym(A).

Prove that the function

ρ: Iso(R2 , A) → Sym(A)

given by ρ(?) = ?|A  is a homomorphism. [4 marks]

(c) Let a, b, c ∈ R2 denote three distinct points in the plane which do not lie on a straight line. That is, a, b, c form a triangle. Let ? be an isometry which fifixes the triangle ?abc, that is ?(a) = a and ?(b) = b and ?(c) = c. Prove that ?(x) = x for any x ∈ R2 , namely ? is the identity transformation.

You may assume that distinct circles in R2 have at most 2 intersection points. [5 marks]

(d) Let A be the equilateral triangle in the plane with vertices T = {1, e2πi/3 , e4πi/3} ⊆ C. It should be clear that Iso(R2 , A) = Iso(R2 , T). (Convince yourself that this is the case). By labelling the vertices of A (i.e the elements of T) with the letters 1, 2, 3 we obtain a homomorphism

ρ: Iso(R2 , T) → S3.

Show that its kernel is trivial. Is ρ injective?

List all the elements of S3 and for each one of them check if it is obtained from an isometry of the plane (for example (12) is obtained by reflflecting the plane at the line through the vertex labelled 3 and the midpoint of the edge formed by vertex 1 and 2).

Deduce that ρ is bijective, hence an isomorphism. We have shown that the group of isometries of the triangle is isomorphic to S3. [8 marks]

(e) Now let A be the square in R2 with vertices T = {(±1, ±1)}. For convenience labelthe vertices by 1, 2, 3, 4. As above, convince yourself that Iso(R2 , A) = Iso(R2 , T). We obtain a homomorphism

ρ: Iso(R2 , T) → S4.

Prove that ρ is injective. Go through all the 24 elements of S4 and check which ones of them is in the image of ρ, namely which permutations of the vertices T arise from an isometry of the plane (you should end up with 8 elements of S4). Deduce that the group of isometries of the square is isomorphic to a subgroup of order 8 of S4. It is called the dihedral group of order 8 and denoted D8. [5 marks]