UDC 519.44

MATHEMATICS

E. M. LEVICH

ON THE REPRESENTATION OF SOLVABLE GROUPS BY MATRICES OVER A CERTAIN FIELD OF CHARACTERISTIC ZERO

(Presented by Academician V. M. Glushkov, 18 II 1969)

One of the basic results in the theory of finite-dimensional representations of solvable groups is the theorem of A. I. Mal'tsev \((^1)\): a solvable matrix group has a normal divisor of finite index whose commutant is nilpotent. In connection with this theorem the question arises: what abstract properties must a solvable group possess in order that it can be represented isomorphically by matrices over some field? As an example one may cite the following assertion: every polycyclic group is representable by matrices over the field of rational numbers \((^2,{}^3)\).

We give the definitions needed in what follows. Let \(G\) be an arbitrary locally nilpotent torsion-free group; \(\Omega\) an arbitrary field of characteristic zero, and \(x^\lambda\) a single-valued function which assigns to any elements \(x \in G\) and \(\lambda \in \Omega\) a certain element of \(G\).

\(G\) is called an \(\Omega\)-powered group \((^4)\) if the following conditions are satisfied:

1) \(x^1 = x,\quad x^{\lambda+\mu} = x^\lambda x^\mu,\quad x^{\lambda\mu} = (x^\lambda)^\mu;\)

2) \(y^{-1}x^\lambda y = (y^{-1}xy)^\lambda,\)

3)
\[ x_1^\lambda x_2^\lambda \cdots x_n^\lambda = t_1^\lambda t_2^{\binom{\lambda}{2}}\cdots t_k^{\binom{\lambda}{c}}, \]
where \(c\) is the nilpotency class of the group generated by the elements \(x_1, x_2, \ldots, x_n\), and \(t_1, t_2, \ldots, t_c\) are Petresco words \((^4)\), p. 21).

In condition 1), \(1\) is the identity of the field \(\Omega\); the elements \(x_i, x, y\) are arbitrary in \(G\), and \(\lambda, \mu\) are arbitrary elements of \(\Omega\); finally, condition 3) is assumed to hold for every finite \(n\).

An \(\Omega\)-powered locally nilpotent group \(G\) is called an \(\Omega R\)-powered group if from the equality \(x^\lambda = y^\lambda\) for some \(\lambda \in \Omega\) it follows that \(x = y\). If \(n\) is the least number which bounds from above the minimal number of \(\Omega\)-generators of each finitely generated \(\Omega\)-powered subgroup of the \(\Omega\)-powered group \(G\), then we shall say that \(G\) has \(\Omega\)-rank \(n\). The notion of an \(\Omega\)-homomorphism of one \(\Omega\)-powered group into another is defined in the natural way.

Theorem 1. In order that a finitely generated solvable torsion-free group \(\Gamma\) have a faithful matrix representation over a field \(\Omega\) of characteristic zero, it is necessary and sufficient that the group \(\Gamma\) have the structure
\[ \Gamma \supset \Gamma_1 \supset \Gamma_2 \supset \{e\}, \]
where \(\Gamma/\Gamma_1\) is a finite group, \(\Gamma_1/\Gamma_2\) is a finitely generated abelian group, and the group \(\Gamma_2\) can be embedded isomorphically in an \(\Omega R\)-powered nilpotent group \(H\) of finite \(\Omega\)-rank, moreover the restriction of each inner automorphism of the group \(\Gamma_1\) to the subgroup \(\Gamma_2\) induces an \(\Omega\)-automorphism of the group \(H\).

Corollary. In order that a finitely generated solvable torsion-free group \(\Gamma\) have a faithful matrix representation over the field of ra

rational numbers, it is necessary and sufficient that the group \(\Gamma\) have a series

\[ \Gamma \supset \Gamma_1 \supset \Gamma_2 \supset \{e\}, \]

where \(\Gamma/\Gamma_1\) is a finite group, \(\Gamma_1/\Gamma_2\) is a finitely generated abelian group, and \(\Gamma_2\) is a torsion-free nilpotent group of finite rational rank.

This corollary gives an answer to the question posed in \((^5)\).

Theorem 2. In order that a torsion-free solvable group \(\Gamma\) with trivial center be isomorphically represented by matrices over some field \(\Omega\) of characteristic zero, it is necessary and sufficient that the following conditions be satisfied:

1) in \(\Gamma\) the minimal condition holds for the centralizers of an ascending sequence of subgroups of \(\Gamma\);

2) in \(\Gamma\) there is a normal series

\[ \Gamma \supset \Gamma_1 \supset \Gamma_2 \supset \{e\}, \]

where \(\Gamma/\Gamma_1\) is a finite group, \(\Gamma_1/\Gamma_2\) is an abelian group, and the group \(\Gamma_2\) can be isomorphically embedded in the \(\Omega R\)-completion of a nilpotent group \(H\) of finite \(\Omega\)-rank, with the restriction of each inner automorphism of the group \(\Gamma_1\) to the subgroup \(\Gamma_2\) inducing an \(\Omega\)-automorphism of the group \(H\).

Using the theorem of A. I. Mal’cev cited above, D. M. Smirnov showed in \((^6)\) that a finitely generated free solvable group of derived length 3 has no faithful matrix representation over any field. In connection with this there arises the question of the representability of two-step solvable groups by matrices over some field of characteristic zero. It is quite easy to construct an example of a two-step solvable torsion-free group which cannot be faithfully represented by matrices over any field of characteristic zero. In particular, such a group will be the group \(\Gamma\) which is the semidirect product of the direct sum \(H\) of a countable number of rational groups \(H_n\) \((n=1,2,\ldots)\) and the infinite cyclic group \(\{z\}\), where \(z^{-1}h_n z = n h_n\) \((h_n \in H_n)\).

Theorem 3. Every finitely generated two-step solvable torsion-free group \(\Gamma\) has a faithful matrix representation over some field of characteristic zero.

Theorem 4. The discrete wreath product \(G=\Gamma \operatorname{wr} H\), where \(\Gamma\) and \(H\) are torsion-free abelian groups, is isomorphically embeddable in the group of matrices of the second order over some field of characteristic zero.

Theorem 5*. A free nilpotent group of nilpotency class \(n\) admits a faithful matrix representation over some field of characteristic zero.

Latvian State University
named after P. Stučka
Riga

Received
10 II 1969

CITED LITERATURE

\(^1\) A. I. Mal’cev, Matem. sborn., 28, 567 (1951).
\(^2\) L. Auslander, Ann. Math., No. 7 (1967).
\(^3\) R. Swan, Proc. Am. Math. Soc., 18, 385 (1967).
\(^4\) F. Hall, Collection of Translations. Mathematics, 12, 1, 3 (1968).
\(^5\) M. I. Kargapolov, Algebra and Logic, Seminar, 6, 5, 17 (1967).
\(^6\) D. M. Smirnov, DAN, 155, No. 3, 535 (1964).

* This theorem was obtained jointly with V. G. Vilyatser.