The Automorphism Groups of Generalized McLain Groups

Abstract

In 1954, McLain [16] applied the well-known connection between nilpotent algebras (of triangular matrices) and nilpotent groups to establish the existence of particular locally nilpotent groups, now known as McLain groups. McLain groups are important for many questions in group theory, and their construction and properties can be found in most text books of infinite group theory; we refer to [17, pp. 347–349]. The basic ingredients in McLain’s construction are the linear ordering ℚ of rational numbers and the field F p of p elements (p a prime), which give rise to McLain groups G(F p ,ℚ) which consist of upper triangular “ℚ x ℚ-matrices” over F p (rows and columns labeled by numbers in ℚ) with 1’s on the diagonal and only finitely many entries from F p \{0} elsewhere. These matrices are clearly invertible and constitute a group G = G(F p ,ℚ). The field F p makes G a locally finite p-group and even if F p is replaced by a field of characteristic 0, matrix multiplication ensures that G is locally nilpotent; moreover ℚ is homogeneous, hence G is characteristically simple — in contrast to finite p-group. McLain’s ingenious, but also very transparent construction stimulated further research on McLain groups and their natural extensions. Most notably, J. Roseblade deeply investigated in his Ph. D. thesis under supervision of P. Hall in 1963 the “internal structure” of G and determined its automorphism group; see [18]. Moreover, Wilson [19] studied generalized McLain groups G(F p ,S) replacing (ℚ, ≤) by another dense linear ordering S. His extensions of McLain’s work are a source for many interesting examples which illustrate the strength and limitation of recent results in group theory.