Group Theory: Math30038, Sheet 9

GCS

1. Suppose that G is a simple group of order 60.

(a) Show that G has a subgroup A of order 12.

(b) Show that A has exactly 5 different conjugates.

(c) Show that there is an injective homomorphism from G to S_{5}. Let the image of this map be H, a subgroup of index 2 in S_{5}.

(d) Show that both A_{5} and H both contain every element of S_{5} of the form g^{2} and therefore every 5-cycle and every 3-cycle.

(e) Show that H = A_{5}.

(f) Deduce that any simple group of order 60 must be isomorphic to

A_{5}.

2. Show that the following presentations all describe the trivial group. (a) ⟨x | x^{2}, x^{3}⟩,

(b) ⟨x, y | xy = yx, xyx = yxy, x^{5}, y^{6}⟩ and

(c) ⟨x, y | x^{2}, y^{4}, xyx = y^{2}, xyxyxy⟩.

3. We discuss the group G = ⟨x, y | x^{2} = y^{2}, (xy)^{2} = x^{2}, x^{4}⟩ in stages: (a) Let T = ⟨x^{2}⟩ ≤ G so |T | ≤ 2. Show that T ⊴ G.

(b) Let H = G/T. Show that xT and yT commute so that H is abelian.

(c) Show that |H| ≤ 4.

(d) Deduce that |G| ≤ 8.

(e) Find a group of order 8 containing elements a, b such that a^{2} = b^{2}, (ab)^{2} = a^{2} and a^{4} = 1. Apply von Dyck’s theorem to deduce that G has a homomorphic image of order 8, and by part (d) deduce that this homomorphism is an isomorphism.

4. Discuss the (language) group L presented by the small roman alphabet with relations consisting of all homonyms. Thus we have relations such as too = two, practice = practise, sea = see and so on. The group you get in my ideolect is C_{2} (South London with a transparent veneer of civilization), whereas in South African English you get the trivial group. On the other hand, I understand that in German the spelling of a word is determined by its pronunciation, so L is a free group. To get you started in L for English, since two = too in L, then t^{−1}twoo^{−1} = t^{−1}tooo^{−1} and thus w = o in L. Also too = to so o = 1 and therefore w = 1. Here 1 denotes the identity element of L for English.