THIN GROUPS OF PRIME-POWER ORDER AND THIN LIE ALGEBRAS

Abstract

IN this paper we extend the theory of thin groups of prime-power order in a direction suggested by the theory of groups of prime-power order with fixed co-class, [9,14]. Moreover, we have found it helpful to use associated Lie algebras in this study. Let p be a prime. Recall that a group of p-power order is thin [2] if all the anti-chains in its lattice of normal subgroups are short. Here an anti-chain in a lattice is a subset of pairwise incomparable elements; and by saying that anti-chains are short we mean that they have length at most p +1. In fact, the definition extends to pro-/? -groups, and their lattices of (closed) normal subgroups. Thus p -groups of maximal class, or co-class 1, are examples of thin groups; for the theory of groups of co-class 1 see [10]. Examples of thin infinite pro-p-groups are the binary p-adic group, [7, 111.17], and the Nottingham group [9,15]. Brandl, Caranti and Scoppola [3] have shown that metabelian thin p-groups have order at most p?. It is clear that the lattice of normal subgroups of a non-(pro-)cyclic (pro-)p-group has anti-chains of length p +1 . To avoid trivialities, it is convenient to exclude (pro-)cyclic groups from our consideration of thin groups here. The only thin abelian p-group is thus elementary of order p; its lattice of normal subgroups we refer to as a diamond. It follows that the commutator factor group of a thin group is elementary of order p, so that thin groups are 2-generator groups, and every factor of the lower central series has exponent p and order at most p. It also follows (see [2,3] and the next section) that every non-trivial normal (closed) subgroup in a thin group lies between two consecutive terms of the lower central series so that in a thin pro-p-group all normal subgroups are of finite index, that is, open. Thus the lattice of normal subgroups of a thin group looks like a sequence of diamonds linked by chains. The groups of