MATHEMATICS

A. L. SHMEL'KIN

ON FREE POLYNILPOTENT GROUPS

(Presented by Academician A. I. Mal'cev on 24 XI 1965)

We shall use the following notation: \(G_s\) is the \(s\)-th term of the lower central series of the group \(G\), defined inductively: \(G_s=[G_{s-1},G]\), if \(s>1\), and \(G_1=G\); if a sequence \(n_1,n_2,\ldots,n_k,\ldots\) of positive integers is given, then \(G_{n_1,\ldots,n_k}\) denotes the subgroup, also defined inductively: \(G_{n_1,\ldots,n_k}=(G_{n_1,\ldots,n_{k-1}})_{n_k}\), the \(n_k\)-th term of the lower central series of the group \(G_{n_1,\ldots,n_{k-1}}\), if \(k>1\).

If \(G_{n_1,\ldots,n_k}=E\), then the group \(G\) is called polynilpotent with respect to the sequence \(n_1,\ldots,n_k\); all such groups form the variety \(\mathfrak{P}\) of polynilpotent groups with respect to the sequence \(n_1,\ldots,n_k\), which is represented in the form of the product

\[ \mathfrak{P}=\mathfrak{N}_{n_k-1}\mathfrak{N}_{n_{k-1}-1}\cdots\mathfrak{N}_{n_1-1}; \]

here \(\mathfrak{N}_c\) denotes the variety of all nilpotent groups of class \(\leq c\).

The free groups of the variety \(\mathfrak{P}\) are called free polynilpotent groups. A free polynilpotent group \(G\) is represented in the form of a factor group \(G\cong F/F_{n_1,\ldots,n_k}\) of a free group \(F\) of the same rank as the group \(G\). In what follows, by \(G\) we shall denote a free polynilpotent group of arbitrary rank with respect to the sequence \(n_1,\ldots,n_k\).

Free polynilpotent groups have been studied in a number of works (see \((^{1-5})\)). The present note is a natural continuation of the author’s investigations \((^{2,3})\).

In \((^{2,3})\) the \(W\)-free subgroups of free soluble groups were completely described, where \(W\) is an arbitrary variety (we recall that free soluble groups are a special case of free polynilpotent groups when \(n_1=\cdots=n_k=2\)). Here we formulate a theorem describing the \(W\)-free subgroups of free polynilpotent groups, from which the theorem on free soluble groups follows directly.

Theorem 1. Let \(H\) be a subgroup of the group \(G\) such that \(H\subseteq G_{n_1,\ldots,n_s}\), but \(H\nsubseteq G_{n_1,\ldots,n_{s+1}}\). If \(H\) is \(W\)-free with respect to some variety \(W\), then \(H\) is a free polynilpotent group. The subgroup \(H\) is a free polynilpotent group if and only if it has a system of generators \(h_i,\ i\in I\), which modulo \(G_{n_1,\ldots,n_{s+1}}\) freely generate a free nilpotent group. If the class of this free nilpotent group is equal to \(c-1\) \((c\leq n_{s+1})\), then \(H\) corresponds to the sequence \(c,n_{s+2},\ldots,n_k\).

In the proof of this theorem, besides the known properties of free polynilpotent groups, the following lemmas are used.

Lemma 1. A \(W\)-free subgroup of a free nilpotent group is free nilpotent.

Lemma 2. Let \(F/N\) be a free nilpotent group of finite rank \(r\), where \(F\) is a free group. Then in \(F\) one can choose a system of free generators \(x_1,\ldots,x_r,\ldots\) such that the images of the elements \(x_1,\ldots,x_r\) in \(F/N\) are free generators of the group \(F/N\).

The following theorems were also unknown for free solvable groups.

Theorem 2. Let \(H\) be such a free polynilpotent subgroup of the group \(G\) that
\[ H \supseteq G_{n_1,\ldots,n_{k-1}}, \]
then \(H\) coincides with one of the subgroups \(G_{n_1,\ldots,n_s}\), \(s=1,\ldots,k-1\).

Theorem 3. Let \(H\) be such a free polynilpotent subgroup of the group \(G\) that
\[ H \supseteq G_{n_1,\ldots,n_{k-1},\,n_k-1}; \]
then \(H\) either coincides with one of the subgroups \(G_{n_1,\ldots,n_s}\), \(s=1,\ldots,k-1\), or \(H\) is abelian.

Finally, the following theorem completely describes the free polynilpotent normal divisors of the group \(G\).

Theorem 4. Let \(H\) be an invariant free polynilpotent subgroup of the group \(G\). Then either \(H\) coincides with one of the subgroups \(G_{n_1,\ldots,n_s}\), \(s=1,\ldots,k-1\), or \(H\) is free abelian and lies in \(G_{n_1,\ldots,n_{k-1}}\); in the latter case, if \(k\ge 2\), then
\[ H \subseteq G_{n_1,\ldots,n_{k-1},\,l}, \]
where
\[ l=\left[\frac{n_k+1}{2}\right]. \]

Corollary. Every free nilpotent normal divisor of a free nilpotent group either coincides with the whole group, or is abelian.

Let us also note that from the results of the work \({}^{(6)}\) it can be deduced that the set of nonisomorphic three-step solvable groups with two generators has the cardinality of the continuum. This shows that free polynilpotent groups are very rich in normal divisors.

Moscow State University
named after M. V. Lomonosov

Received
28 IX 1965

CITED LITERATURE

\({}^{1}\) K. Gruenberg, Proc. London Math. Soc., 7, No. 25, 29 (1957).
\({}^{2}\) A. L. Shmelkin, DAN, 151, No. 1, 73 (1963).
\({}^{3}\) A. L. Shmelkin, Izv. AN SSSR, ser. matem., 28, No. 1, 91 (1964).
\({}^{4}\) G. Baumslag, Trans. Am. Math. Soc., 108, 516 (1963).
\({}^{5}\) G. Baumslag, B. H. Neumann et al., Math. Zs., 86, No. 1, 93 (1964).
\({}^{6}\) B. H. Neumann, H. Neumann, J. London Math. Soc. 34, No. 4, 465 (1959).