Classical Groups in Dimension 3 as Completions of the Goldschmidt G 3 ‐Amalgam

The aim of this paper is to determine which (finite) 3‐dimensional classical groups are completions of the Goldschmidt G 3 ‐amalgam. We recall, first, that an amalgam (of rank 2) consists of three groups P 1 , P 2 , B and two group monomorphisms ϕ 1 , ϕ 2 such that ϕ 1 : B → P 1 and ϕ 2 : B → P 2 Usually, when ϕ 1 and ϕ 2 are understood, this amalgam is denoted by A( P 1 , P 2 , B ). Now a group G is a completion of A( P 1 , P 2 , B ) if there exist group homomorphisms ψ i : P i → G ( i = 1, 2) satisfying G = 〈im ψ 1 , im ψ 2 〉 and ψ 1 ϕ 1 = ψ 2 ϕ 2 : B → G . The Goldschmidt G 3 ‐amalgam, which appears in [ 6 ], is defined as follows: P 1 ≅ Sym(4) ≅ P 2 , B ≅ Dih(8) (Dih( n ) denotes the dihedral group of order n ) withϕ 1 ‐ 1( O 2 ( P 1 )) ≠ϕ 2 ‐ 1( O 2 ( P 2 )).