V. G. ZHITOMIRSKII

TRIANGULAR GROUPS OF AUTOMORPHISMS OF QUASI-OPERATOR GROUPS

(Presented by Academician A. I. Mal’cev, 15 I 1962)

Let (G) be a group and (\Omega) some ring of principal ideals. Suppose that to each element (g \in G) and to each (\omega \in \Omega) there corresponds uniquely an element of the group (G) (we shall denote it by (g^\omega)), and that the following relations hold:

1. (g^{\omega+\nu}=g^\omega g^\nu) for any (g \in G) and (\omega,\nu \in \Omega).

2. (g^{\omega\nu}=(g^\omega)^\nu) for any (g \in G) and (\omega,\nu \in \Omega).

3. For the identity (1) of the ring (\Omega), (g^1=g) for all (g \in G).

4. For the identity element (e) of the group (G) and any (\omega \in \Omega), (e^\omega=e).

5. (x^{-1}g^\omega x=(x^{-1}gx)^\omega) for any (x,g \in G) and (\omega \in \Omega).

A subgroup (H \subset G), as usual, is called (\Omega)-admissible if, for any (h \in H) and (\omega \in \Omega), (h^\omega \in H). The intersection of any set of (\Omega)-admissible subgroups is (\Omega)-admissible. Therefore it makes sense to speak of the (\Omega)-closure of a set of elements (M \subset G). The following relation connects the operation in (G) with the action of the elements of (\Omega).

1. For any (a,b \in G) and (\omega \in \Omega),
   [
   (ab)^\omega=a^\omega b^\omega c,
   ]
   where (c) belongs to the (\Omega)-closure of the commutant of the subgroup generated by the elements (a) and (b):
   [
   c \in (K{a,b})^\Omega.
   ]

If relations 1—6 hold, then we shall call (G) an (\Omega)-group, and the elements of (\Omega) quasi-operators. Let us note that a similar axiomatics, for other purposes, was considered by N. F. Sesekin ((^6)).

We shall call an (\Omega)-group (G) a group without (\Omega)-torsion if from (g^\omega=e) it follows that (g=e) for any nonzero element (\omega \in \Omega). A group without (\Omega)-torsion will be called an (\Omega R)-group if, for any (nonzero) (\omega \in \Omega), from (a^\omega=b^\omega) it follows that (a=b). An (\Omega)-admissible subgroup (H) of an (\Omega R)-group (G) will be called (\Omega)-isolated if from (g^\omega \in H) it follows that (g \in H). The intersection of any set of (\Omega)-isolated subgroups is (\Omega)-isolated. As usual, the (\Omega)-isolator (I(M)) of a set (M) of elements of an (\Omega R)-group (G) will mean the intersection of all (\Omega)-isolated subgroups containing the set (M).

The concept of an (\Omega R)-group is a generalization of the concept of an (R)-group introduced by P. G. Kontorovich ((^1)). Ordinary (R)-groups are (\Omega R)-groups with the ring of integers as (\Omega). A vector space over a field of characteristic zero is a commutative (\Omega R)-group. An example of a noncommutative (\Omega R)-group is the well-known group of real numbers with multiplication law
[
(x_1,y_1)(x_2,y_2)=(x_1+x_2e^{-y_1},\, y_1+y_2),
]
if, for any real (\omega), one sets
[
(x,y)^\omega=\left(x\frac{1-e^{-\omega y}}{1-e^{-y}},\, \omega y\right).
]

Many facts from the theory of (R)-groups and, in particular, from the theory of locally nilpotent groups without torsion ((^{1-4})), carry over to (\Omega R)-groups. We shall give some of them.

In an (\Omega R)-group, the centralizer of any set of elements and, consequently, the center of an (\Omega R)-group is an (\Omega)-isolated subgroup.

A group without (\Omega)-torsion is an (\Omega R)-group if and only if its factor group by its center is an (\Omega R)-group.

The set of elements of the form (g^\omega), for fixed (g) and all possible (\omega \in \Omega), is an ((\Omega)-admissible) subgroup, which we shall call (\Omega)-cyclic. If, in an ((\Omega)-admissible) subgroup, every finite set of elements belongs to some (\Omega)-cyclic subgroup, then such a subgroup is naturally called (\Omega)-locally cyclic.

In an (\Omega R)-group the (\Omega)-isolator (I(g)) of any element (g) is an (\Omega)-locally cyclic group, and the (\Omega)-isolators of distinct elements either do not intersect or coincide.

Every locally nilpotent (\Omega)-group without (\Omega)-torsion is an (\Omega R)-group.

In a nilpotent (\Omega R)-group the normalizer of an (\Omega)-isolated subgroup is (\Omega)-isolated.

If (G) is a locally nilpotent (\Omega R)-group, then every element (a) of the (\Omega)-isolator (I(H)) of an arbitrary (\Omega)-admissible subgroup (H) has the property that, for some (\omega \in \Omega), (a^\omega \in H).

If (G) is an (\Omega R)-group and (H) is its subgroup possessing an ascending central series, then the (\Omega)-isolator (I(H)) also possesses an ascending central series, and the classes of (H) and (I(H)) are one and the same.

In an (\Omega R)-group the (\Omega)-isolator of a locally nilpotent subgroup is locally nilpotent.

The locally nilpotent radical (R(G)) of an (\Omega R)-group (G) is (\Omega)-isolated.

The proofs of these facts are, in the main, analogous to the proofs of the corresponding facts in the theory of (R)-groups. The principal changes are connected with the necessity of checking the (\Omega)-admissibility of the subgroups under consideration and with considering the properties of (\Omega)-closures.

An automorphism (\varphi) of an (\Omega)-group (G) will be called an (\Omega)-automorphism if, for all (g \in G) and (\omega \in \Omega), (\varphi(g^\omega)=\varphi(g)^\omega). A group (\Phi) of (\Omega)-automorphisms of an (\Omega R)-group (G) will be called scalar if the group (G) is covered by (\Phi)-admissible (\Omega)-locally cyclic subgroups.

A scalar group of automorphisms is abelian.

If in an (\Omega R)-group (G) there is a normal series consisting of (\Phi)-admissible (\Omega)-isolated subgroups such that in every factor of this series (\Phi) induces a scalar group of automorphisms, then (\Phi) will be called a triangular group of automorphisms. Together with the group of (\Omega)-automorphisms (\Phi) we shall consider the group (\overline{\Phi}), equal to the product of (\Phi) and the group (\hat G) of inner automorphisms of the group (G).

Theorem. Let (G) be an (\Omega R)-group with the maximality condition for (\Omega)-isolated subgroups, and let (\Phi) be a group of its (\Omega)-automorphisms. Suppose also that (\overline{\Phi}) is triangular relative to (R(G)) and (\Phi) is triangular relative to the factor group (G/R(G)).

Then:

1. The factor group (\Phi/R(\Phi,G)) of the group (\Phi) by the locally stable radical (R(\Phi,G)) ({}^{(5)}) is abelian.
2. (R(\Phi,G)) is a nilpotent group.
3. In (R(\Phi,G)) there is a series invariant under (\Phi), the factors of which are isomorphic to subgroups of the factors of a triangular series in (R(G)).

In proving the theorem the following proposition is used essentially.

Let (G) be an (\Omega R)-group and let (H) be its normal divisor that is a maximal (\Omega)-isolated subgroup such that the group (\Psi) of (\Omega)-automorphisms of the group (G) induces the identity automorphism in (H) and in the factor group (G/H). Suppose, moreover, that in (H) there is given a series invariant in (G) with (\Omega)-locally cyclic factors
[
E = H_0 \subset H_1 \subset \cdots \subset H_i \subset \cdots \subset H_k \subset H.
]
Form in (\Psi) the series
[
E = \Psi_0 \subset \Psi_1 \subset \cdots \subset \Psi_i \subset \cdots \subset \Psi_k \subset \Psi,
]
where (\Psi_i) is the (\Psi)-centralizer of the factor group (G/H_i).

Then:

1. The series in (\Psi) is invariant under every containing group of automorphisms (\Psi), relative to which the series in (H) is admissible.
2. The factors of the series in (\Psi) are isomorphic to subgroups of the factors of the series in (H).

We shall present the main stages of the proof of the theorem. The following is proved. A (\overline{\Phi})-triangular series in (R(G)) is finite, has (\Omega)-locally cyclic factors, and is invariant under (\hat G). A (\Phi)-triangular series in (G/R(G)) is finite. Therefore

the triangular series in the group (G) is finite. The locally stable radical (R(\Phi, G)) coincides with the (\Phi)-centralizer of the triangular series in (G) and, consequently, is nilpotent. The factor group (\Phi/R(\Phi,G)), being a subdirect product of abelian groups, is itself abelian. The series in (R(\Phi,G)) indicated in the theorem is constructed on the basis of the proposition given above, if one takes into account that (R(\Phi,G)) induces the identity automorphism in (G/R(G)) and that in the group (G) the maximality condition holds for (\Omega)-isolated subgroups.

Let us apply the results obtained to the study of groups preserving the order of automorphisms of an ordered (\Omega)-group.

Consider an (\Omega)-group (G), for which (\Omega) is an ordered field. In the case when (\Omega) is a field, every (\Omega)-group is, obviously, an (\Omega R)-group, and in it the notions of (\Omega)-admissibility and (\Omega)-isolation coincide. We shall call such a group (G) an ordered (\Omega)-group if (G) is an ordered group and 1) from (a>b) it follows that (a^\omega>b^\omega) for every positive element of the field; 2) from (a>e) and (\omega_1>\omega_2) it follows that (a^{\omega_1}>a^{\omega_2}).

In an ordered (\Omega)-group every convex subgroup is (\Omega)-isolated. Let (G) be such an ordered (\Omega)-group that in it there is a well-ordered increasing series of convex subgroups

[
E=G_0 \subset G_1 \subset \ldots \subset G_\gamma \subset G_{\gamma+1} \subset \ldots
\tag{1}
]

and the factors of this series are (\Omega)-cyclic groups. It is not difficult to show that (G) will in this case be a (ZA)-group. Consider the group (\Phi) of all order-preserving (\Omega)-automorphisms of the group (G). The series (1) will be (\Phi)-admissible and, consequently, (\Phi) is a triangular group of automorphisms. The group of automorphisms (\Phi) can be ordered.

We give the main steps of the proof. It is not difficult to show that each convex subgroup (G_\gamma) is distinguished in (G_{\gamma+1}) by a semidirect factor: (G_{\gamma+1}=G_\gamma \lambda B_{\gamma+1}), where (B_{\gamma+1}) is an (\Omega)-cyclic group. We shall say that an automorphism (\varphi \in \Phi) belongs to the subgroup (G_{\gamma+1}) if (G_{\gamma+1}) is the first of the subgroups in the series (1) on which the automorphism (\varphi) acts non-identically. If the automorphism (\varphi) belongs to the subgroup (G_{\gamma+1}), then for all positive elements (g) such that (g\in G_{\gamma+1}), but (g\notin G_\gamma), at the same time either (\varphi(g)>g), or (\varphi(g)\varepsilon), where (\varepsilon) is the identity automorphism), and in the second—negative. It is not difficult to verify that the group (\Phi) is ordered in this way.

Denote by (\Sigma) the (\Phi)-centralizer of the series (1). (\Sigma) is an invariant subgroup in (\Phi). Of two automorphisms (\sigma,\psi\in\Sigma), the greater is the one which belongs to the subgroup with the smaller index. If, however, (\sigma) and (\psi) belong to the subgroup (G_{\gamma+1}), then the greater of them will be the one which induces the identity automorphism in the factor group (G_{\gamma+1}/G_\beta) ((\beta\leqslant\gamma)) with the greater index (\beta). From what has been said it follows that the subgroup (\Sigma_{\gamma+1}^{\beta}) of (\Sigma), consisting of all those automorphisms which induce the identity automorphism in (G_\gamma) and (G_{\gamma+1}/G_\beta), is convex in (\Sigma). All such possible subgroups form a convex series in (\Sigma). If the series (1) is finite, then on the basis of the theorem one may assert that the factors of the convex series in (\Sigma) obtained in this way are isomorphic to subgroups of the factors of the series (1).

Received
9 I 1962

CITED LITERATURE

1. P. G. Kontorovich, Matem. sborn., 22, 79 (1948).
2. A. G. Kurosh, Theory of Groups, Moscow, 1953.
3. A. I. Mal’tsev, Izv. AN SSSR, ser. matem., 13, 201 (1949).
4. B. I. Plotkin, Matem. sborn., 30, 197 (1952).
5. B. I. Plotkin, DAN, 140, No. 5, 1019 (1961).
6. N. F. Sesekin, Tr. III Vsesoyuzn. matem. s”ezda, 2, Moscow, 1956, p. 114.