The Macdonald group

Abstract

Given α ∈ Z, the Macdonald group G(α) is defined by G(α) = ⟨ A,B | A[A,B] = Aα, B[B,A] = Bα ⟩. It is known that G(α) is finite if and only if α ̸= 1, in which case the prime factors of |G(α)| are those of α − 1. It is also known that G(α) is nilpotent in certain cases. We show that G(α) is always nilpotent, so that for α ̸= 1, G(α) is the direct product of its Sylow subgroups. In the first third of the thesis, we determine the order, upper and lower central series, nilpotency class, and exponent of each of these Sylow subgroups. For the remaining two thirds of the thesis we concentrate on the Sylow 2-subgroup J = J(α) of G(α), so we assume that α = 1 + 2mℓ, where m ≥ 1 and ℓ is odd. We show that J has presentation J = ⟨ x, y | x[x,y] = x1+2mℓ, y[y,x] = y1+2mℓ, x23m−1 = 1 = y23m−1⟩, order 27m−3, and nilpotency class 5 if m > 1 and 3 if m = 1. In the middle third of the thesis, we determine the automorphism groups of the 2-groups J, H = J/Z(J) and K = H/Z(H), where |H| = 26m−3 and |K| = 25m−3. Explicit multiplication, power, and commutator formulas for J, H, and K are given, and used in the calculation of Aut(J), Aut(H), and Aut(K). In the final third of the thesis, we consider the infinite family of finite 2-groups {J(α)}α̸=1 and settle the following isomorphism problem: given α ̸= 1 ̸= α′ ∈ Z, when are J(α) and J(α′) isomorphic?