UDC 519.41/47

MATHEMATICS

D. I. ZAĬTSEV

ON SOLVABLE GROUPS OF FINITE RANK

(Presented by Academician V. M. Glushkov on 16 X 1967)

A rational series of a group \(\mathfrak{G}\) is an ascending normal series of this group
\[ \mathfrak{R}_0=E\subset \mathfrak{R}_1\subset \ldots \subset \mathfrak{R}_\alpha \subset \mathfrak{R}_{\alpha+1}\subset \ldots \subset \mathfrak{R}_\gamma=\mathfrak{G}, \]
whose factors \(\mathfrak{R}_{\alpha+1}/\mathfrak{R}_\alpha\) are isomorphic to certain subgroups of the additive group of rational numbers \((^1)\).

For a group \(\mathfrak{G}\) possessing a rational series, the notion of rational rank is introduced. Namely, if \(\mathfrak{G}\) has a rational series of finite length \(k\), then the rational rank of \(\mathfrak{G}\) is taken to be \(k\); otherwise the rational rank of the group is considered infinite. We shall denote the rational rank of the group \(\mathfrak{G}\) by \(r(\mathfrak{G})\). The symbol \(s(\mathfrak{G})\) denotes the special rank \((^2)\) of the group \(\mathfrak{G}\).

The question naturally arises of the relation between \(r(\mathfrak{G})\) and \(s(\mathfrak{G})\). In this direction V. M. Glushkov \((^3)\) established that in the case of a locally nilpotent group \(\mathfrak{G}\) the equality \(r(\mathfrak{G})=s(\mathfrak{G})\) holds. In the present article it is reported (Theorem 2) that, for any group possessing a rational series, the special rank is equal to the rational rank.

The proof of the main result is based on a number of properties of solvable groups satisfying the weak minimal condition for subgroups. Let us recall the definition.

A group \(\mathfrak{G}\) satisfies the weak minimal condition for subgroups if there does not exist in it an infinite descending chain of subgroups
\[ \mathfrak{G}_1 \supset \mathfrak{G}_2 \supset \ldots \supset \mathfrak{G}_k \supset \mathfrak{G}_{k+1}\supset \ldots, \]
satisfying the following condition: the index \([\mathfrak{G}_k:\mathfrak{G}_{k+1}]\) of the subgroup \(\mathfrak{G}_{k+1}\) in the subgroup \(\mathfrak{G}_k\) is infinite, \(k=1,2,\ldots\). Groups satisfying this minimal condition were studied by the author in \((^4)\).

We now formulate an auxiliary proposition, which gives a condition that singles out, from the class of nilpotent groups of finite rank, the class of nilpotent groups satisfying the weak minimal condition for subgroups.

Lemma 1. Let there exist in the nilpotent group of finite rank \(\mathfrak{G}\) a system of generating elements \(X_1, X_2, \ldots, X_k\), \(\mathfrak{M}\), and such a natural number \(n\) that for every element \(Y \in \mathfrak{M}\) there is an \(l\) for which
\[ Y^{n^l}\in \{X_1,X_2,\ldots,X_k\}. \]
Then \(\mathfrak{G}\) satisfies the weak minimal condition for subgroups, and for every \(G\in \mathfrak{G}\) one can specify a number \(m\) for which
\[ G^{n^m}\in \{X_1,X_2,\ldots,X_k\}. \]

Conversely, if a nilpotent group \(\mathfrak{G}\) satisfies the weak minimal condition for subgroups, then in \(\mathfrak{G}\) there exists such a finite system of elements \(X_1, X_2,\ldots,X_k\) and such a natural number \(n\) that for every element \(G\in \mathfrak{G}\) one can specify a number \(m\) for which \(G^{n^m}\in \{X_1,X_2,\ldots,X_k\}\).

Corollary 1. In a nilpotent group of finite rank \(\mathfrak{G}\), a subgroup generated by a finite set of subgroups satisfying the weak minimal condition also satisfies the weak minimal condition.

Corollary 2. In a locally nilpotent torsion-free group \(\mathfrak{G}\), a subgroup generated by a finite set of subgroups satisfying the weak minimal condition also satisfies the weak minimal condition.

This follows from Theorem 4 of paper (³) and Corollary 1 of Lemma 1. Let us note that for a periodic locally nilpotent group, Corollary 2 does not hold. A counterexample can be found in the paper of O. Yu. Schmidt (⁵).

One characterization of nilpotent torsion-free groups of finite rank is given by

Lemma 2. A nilpotent torsion-free group \(\mathfrak R\) has finite rank if and only if there does not exist in \(\mathfrak R\) an infinite system of elements \(X_1, X_2, \ldots, X_i, \ldots\) having the following property:

\[ \{X_1, X_2, \ldots, X_i\}\cap\{X_{i+1}\}=E,\qquad i=1,2,\ldots . \]

Lemma 3. Let \(\varphi\) be an automorphism of a nilpotent group of finite rank \(\mathfrak R\), and let \(\mathfrak L\) be a subgroup of \(\mathfrak R\) satisfying the weak minimality condition for subgroups. Then the least subgroup \(\mathfrak L^*\), invariant with respect to \(\varphi\) and containing \(\mathfrak L\), also satisfies the weak minimality condition for subgroups.

Lemma 4. If \(\mathfrak G\) is a nilpotent group of finite rank and the factor group \(\mathfrak G/\mathfrak G'\) of the group \(\mathfrak G\) by its commutator subgroup \(\mathfrak G'\) satisfies the weak minimality condition for subgroups, then \(\mathfrak G\) also satisfies the weak minimality condition for subgroups.

Relying on the formulated lemmas and the results of the work of A. I. Mal’cev (⁶), one can establish the following theorem, which is of definite interest for the theory of solvable groups of finite rank.

Theorem 1. A solvable torsion-free group of finite rank with a finite number of generators satisfies the weak minimality condition for subgroups.

Lemma 5. Let a group \(\mathfrak G\) possess a rational series of length \(k\) and satisfy the weak minimality condition for subgroups. Then in \(\mathfrak G\), for every sufficiently large prime number \(p\), there exist two characteristic subgroups of finite index \(\mathfrak A, \mathfrak B\), such that \(\mathfrak A\supset \mathfrak B\) and the factor group \(\mathfrak A/\mathfrak B\) is an elementary abelian \(p\)-group of order \(p^k\).

From Lemma 5 and Theorem 1 there immediately follows the main result of the present work.

Theorem 2. If \(\mathfrak G\) is a group possessing a rational series, then the special rank of \(\mathfrak G\) is equal to its rational rank, i.e. \(s(\mathfrak G)=r(\mathfrak G)\).

We indicate some known results obtained as consequences of Theorem 2.

Corollary 1. (V. M. Glushkov (³)). In order that a locally nilpotent torsion-free group be a nilpotent group of finite special rank \(k\), it is necessary and sufficient that it possess a rational series of length \(k\).

Corollary 2. (S. N. Chernikov (¹), B. I. Plotkin (⁷)). If a group possesses a finite rational series, then the length of such a series is an invariant of the group. The length of a rational series of a proper isolated subgroup is strictly less than the length of a rational series of the whole group.

Corollary 3. (B. I. Plotkin (⁸)). Let a group \(\mathfrak G\) possess an ascending rational series. In this case, in order that \(\mathfrak G\) have finite rank, it is necessary and sufficient that \(\mathfrak G\) possess finite special rank.

Institute of Mathematics
Academy of Sciences of the Ukrainian SSR

Received
13 X 1967

REFERENCES

¹ S. N. Chernikov, Uch. zap. Ural’skogo gos. univ., vol. 7, 3 (1949).
² A. I. Mal’cev, Matem. sborn., 22, 351 (1948).
³ V. M. Glushkov, Matem. sborn., 30, 79 (1952).
⁴ P. I. Zaitsev, DAN, 178, No. 4 (1968).
⁵ O. Yu. Schmidt, Matem. sborn., 8, 363 (1940).
⁶ A. I. Mal’cev, Matem. sborn., 28, 567 (1951).
⁷ B. I. Plotkin, Matem. sborn., 30, 197 (1952).
⁸ B. I. Plotkin, Tr. Mosk. matem. obshch., 6, 299 (1957).