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MATHEMATICS
D. M. SMIRNOV
ON GENERALIZED SOLVABLE GROUPS AND THEIR GROUP RINGS
(Presented by Academician A. I. Mal'cev on 2 XII 1963)
- A primitive class or variety of groups is a class of all groups satisfying some fixed system of identical relations.
Let a group \(G\) and some collection of words \(S\) in the unknowns \(X=\{x_1,x_2,\ldots\}\) be given. Denote by \(V(S,G)\) the subgroup of \(G\) generated by all values of the words from \(S\) in the group \(G\), i.e.
\[ V(S,G)=\{u(g_1,\ldots,g_n)\}, \]
where \(u(x_1,\ldots,x_n)\in S,\ g_1,\ldots,g_n\in G\). \(V(S,G)\) is called the verbal subgroup \((^1)\) of the group \(G\) generated by \(S\). If the set \(S\) is empty, then we put \(V(S,G)=1\). The verbal subgroup of a group \(G\) determined by a variety of groups \(\mathfrak M\) will be denoted by \(V(\mathfrak M,G)\). If \(F\) is an arbitrary free group, then the factor group \(F/V(\mathfrak M,F)\) is called a free \(\mathfrak M\)-group.
We shall call a variety of groups \(\mathfrak M\) normal if, for every normal divisor \(A\) of any noncommutative free group \(F\), the equality \(V(\mathfrak M,A)=V(\mathfrak M,F)\) implies \(A=F\).
Let some variety of groups \(\mathfrak M\) be given. A normal system of a group \(G\) all of whose factors belong to the variety \(\mathfrak M\) will be called a normal \(\mathfrak M\)-system.
If a normal system \(\mathfrak A=\{A_\alpha\}\) of a group \(G\) contains a normal divisor \(A\) of this group, then the subsystem \(\mathfrak B\) of the system \(\mathfrak A\) consisting of all \(A_\alpha\) that contain \(A\) (including \(A\) itself) shall be called a normal system of the group \(G\) relative to \(A\). The factor groups of the subgroups from \(\mathfrak B\) by the normal divisor \(A\) evidently form a normal system of the group \(G/A\).
Theorem 1. Let \(\mathfrak M\) be a normal variety of groups, \(F\) a noncommutative free group, and \(A\) a normal divisor of the group \(F\) distinct from the identity. If the group \(F\) possesses a normal \(\mathfrak M\)-system \(\mathfrak S=\{F_\alpha\}\) \((1\leq \alpha\leq \gamma,\ F_1=F,\ F_\gamma=A)\) relative to \(A\), then there also exists a normal system \(\mathfrak S^*\) of the group \(F\) relative to \(V(\mathfrak M,A)\), all of whose factors are subgroups of free \(\mathfrak M\)-groups. The system \(\mathfrak S^*\) may be obtained from the system
\[ \mathfrak S'=\{V_\alpha\}\quad (0\leq \alpha\leq \gamma), \]
where \(V_0=F,\ V_\alpha=V(\mathfrak M,F_\alpha)\) \((1\leq \alpha\leq \gamma)\), by supplementing the latter with the intersections of all subgroups of each of its subsystems.
Fix some variety of groups \(\mathfrak M\) and distinguish the following types of normal systems in groups:
\(N_{\mathfrak M}\) — a normal \(\mathfrak M\)-system,
\(I_{\mathfrak M}\) — an invariant \(\mathfrak M\)-system,
\(N_{\mathfrak M}^{*}\) — a normal \(\mathfrak M\)-system, completely ordered by ascending order,
\(I_{\mathfrak M}^{*}\) — an invariant \(\mathfrak M\)-system, completely ordered by ascending order,
\(K_{\mathfrak M}\)—a normal \(\mathfrak M\)-system, well ordered in descending order,
\(R_{\mathfrak M}\)—a finite normal series all of whose factors belong to the variety \(\mathfrak M\).
We shall call a group \(G\) a \(T\)-group \((T=N_{\mathfrak M}, I_{\mathfrak M}, N_{\mathfrak M}^{*}, I_{\mathfrak M}^{*}, K_{\mathfrak M}, R_{\mathfrak M})\) if it has a normal system of type \(T\). In particular, if \(\mathfrak M\) is the variety of all abelian groups, then the classes \(N_{\mathfrak M}\)-, \(I_{\mathfrak M}\)-, \(N_{\mathfrak M}^{*}\)-, \(I_{\mathfrak M}^{*}\)-, \(K_{\mathfrak M}\)-, \(R_{\mathfrak M}\)-groups coincide respectively with the classes of \(RN\)-, \(RI\)-, \(RN^{*}\)-, \(RI^{*}\)-, \(RK\)-, \(R\)-groups \((^{2})\). From Theorem 1 there follows immediately:
Corollary. Let \(\mathfrak M\) be a normal variety of groups. If the factor group \(F/A\) of a noncommutative free group \(F\) by some nonidentity normal divisor \(A\) is a \(T\)-group \((T=N_{\mathfrak M}, I_{\mathfrak M}, N_{\mathfrak M}^{*}, I_{\mathfrak M}^{*}, K_{\mathfrak M}, R_{\mathfrak M})\), then the factor group \(F/V(\mathfrak M,A)\) has a normal system of the same type \(T\), all factors of which are subgroups of free \(\mathfrak M\)-groups.
If \(A,B\) are arbitrary normal divisors of a noncommutative free group \(F\), then, as M. Auslander and Lyndon established \((^{3})\), the equality \([A,A]=[B,B]\) implies \(A=B\). Consequently, the variety of all abelian groups is normal and, in view of Theorem 1, the following is also true:
Theorem 2. If the factor group \(F/A\) of a free group \(F\) by some normal divisor \(A\) is a \(T\)-group \((T=RN, RI, RN^{*}, RI^{*}, RK, R)\), then the factor group \(F/[A,A]\) has a normal system of type \(T\), all factors of which are free abelian groups.
Hence, in view of the results of M. I. Zaitseva \((^{4})\) and A. A. Bovdi \((^{5})\), there follows immediately:
Corollary. If the factor group \(F/A\) of a free group \(F\) by some normal divisor \(A\) is an \(RN\)-group, then the factor group \(F/A^{(n)}\) of the group \(F\) by any \(n\)-th commutant \(A^{(n)}\) of the subgroup \(A\) is right-orderable, and its group ring \(P\left[F/A^{(n)}\right]\) over any field \(P\) contains no zero divisors.
Let a sequence of natural numbers \(m_1,m_2,\ldots\), greater than 1, be given. In every group \(G\) there is a chain of normal divisors
\(G \supset G_{m_1} \supset G_{m_1,m_2} \supset \cdots\), where \(G_{m_1}\) is the \(m_1\)-th member of the lower central series of the group \(G\), \(G_{m_1,m_2}\) is the \(m_2\)-th member of the lower central series of the group \(G_{m_1}\), and so on. The group \(G\) is called polynilpotent of class \(\mathfrak m=(m_1,\ldots,m_k)\) if \(G_{m_1,\ldots,m_k}=1\). The polynilpotent groups of any given class \(\mathfrak m=(m_1,\ldots,m_k)\) form, by a result of B. H. Neumann \((^{6})\), a normal variety. Therefore Theorem 1 is applicable also in this more general situation.
From the work of H. Neumann \((^{7})\) it follows that the variety of groups defined by the identities \(x^p=1,\ xy=yx\) (\(p\) a prime number) is also normal. The identity \(x^n=1\) \((n>1)\) also defines a normal variety of groups.
In the free semigroup of all varieties of groups distinct from the zero and unit varieties, the normal varieties form an isolated free subsemigroup.
2. A group \(G\) is called radical \((^{8})\) if it has a normal system, well ordered in increasing order, with locally nilpotent factors. A group \(G\) which locally has this property is called locally radical. The class of locally radical groups contains, in particular, all locally solvable groups and all \(RN^{*}\)-groups.
Theorem 3. If \(G\) is an arbitrary locally radical torsion-free group, then the following three conditions are equivalent:
- The integral group ring \(Z[G]\) of the group \(G\) is isomorphically representable by matrices over some field.
II. For any free presentation \(G \cong F/A\) of the group \(G\), the factor group \(F/[A,A]\) is isomorphically representable by matrices over some field.
III. \(G\) is a finite extension of an abelian group.
Corollary. If \(G\) is a locally radicable ordered group (or, in particular, any locally nilpotent torsion-free group), then each of conditions I, II in Theorem 3 is equivalent to the condition
III′. The group \(G\) is abelian.
In the proof of Theorem 3 one uses A. I. Mal’cev’s theorem on locally soluble matrix groups (\({}^{9}\), Theorem 1) and the following propositions.
Lemma 1. Let \(A\) be such a normal divisor of the free group \(F\) that \(F/A\) is a torsion-free group. Then every locally nilpotent subgroup \(H\) of the group \(F/[A,A]\) is either contained in the group \(A/[A,A]\), or is isomorphic to some subgroup of the additive group of rational numbers.
The proof of this lemma can be obtained from A. I. Mal’cev’s theorem (\({}^{10}\)) on commuting elements of the group \(F/[A,A]\).
Lemma 2. If the integral group ring \(Z[G]\) of the group \(G=F/A\) (where \(F\) is a free group and \(A\) is some normal divisor of it) is isomorphically representable by matrices over some field \(P\), then the factor group \(F/[A,A]\) also admits a faithful representation by matrices over a purely transcendental extension of the field \(P\).
Lemma 3. If the group \(G\) contains a subgroup \(H\) of finite index whose integral group ring \(Z[H]\) is isomorphically representable by matrices over some field \(P\) of characteristic zero, then the integral group ring \(Z[G]\) of the group \(G\) also admits a faithful representation by matrices over the same field.
Using Theorem 2 and Lemma 3, it is easy to construct an example of a noncommutative right-orderable group \(G\) whose group ring \(Z[G]\) admits a faithful matrix representation.
Theorem 4. The free \(n\)-step soluble group \(G_n\), for \(n \geq 3\), and the integral group ring \(Z[G_2]\) of the free two-step soluble group \(G_2\) are not faithfully representable by matrices for any choice of the representation field. The free two-step soluble group \(G_2\) is isomorphically representable by matrices over some purely transcendental extension of the field of rational numbers.
Novosibirsk State
University
Received
28 XI 1963
CITED LITERATURE
\({}^{1}\) B. H. Neumann, Math. Ann., 114, 506 (1937).
\({}^{2}\) A. G. Kurosh, S. N. Chernikov, UMN, 2, no. 3, 18 (1947).
\({}^{3}\) M. Auslander, R. C. Lyndon, Am. J. Math., 77, 929 (1955).
\({}^{4}\) M. I. Zaitseva, Uch. Zap. Shuisk. Ped. Inst., 6, 205 (1958).
\({}^{5}\) A. A. Bovdi, Sibirsk. Matem. Zhurn., 1, 555 (1960).
\({}^{6}\) B. H. Neumann, Arch. Math., 13, 4 (1962).
\({}^{7}\) H. Neumann, J. reine u. angew. Math., 212, 109 (1963).
\({}^{8}\) B. I. Plotkin, Matem. sborn., 37, 507 (1955).
\({}^{9}\) A. I. Mal’cev, Matem. sborn., 28, 567 (1951).
\({}^{10}\) A. I. Mal’cev, DAN, 130, 495 (1960).