was solved by Jarden and Ritter [2] and Jarden [3]. The most general result proved in [2] is as follows: Let j~ be a class of finite groups that is closed with respect ...
NORMAL AUTOMORPHISMS
OF DISCRETE GROUPS
V. A. Roman'kov
I.
Introduction.
UDC 519.4
Statement of Principal Results
An automorphism ~ of a group G is called normal if it transforms normal subgroups with finite indices in G.
in themselves all the
According to a well-known theorem of Neukirch [1], all the automorphisms of the absolute Galois group G(Q) of the field Q of rational numbers are normal. Papers by Ikeda, Uchida, Komatsu, and Ivasava (see the references in [2]) confirmed the famous conjecture of Neukirch: All automorphisms of the group G(Q) are inner automorphisms. Here, of course, the authors based their arguments on Neukirch's theorem, proving in fact that normal automorphisms of the group G(Q) are inner automorphisms. The problem about the coincidence of the subgroup Aut n G of normal automorphisms with the subgroup I n n G of inner automorphisms for a profinite group G with well-defined properties was solved by Jarden and Ritter [2] and Jarden [3]. The most general result proved in [2] is as follows: Let j~ be a class of finite groups that is closed with respect to subgroups, homomorphic maps, and extensions, and let G be a p r o - S T - g r o u p given by n generators and k defining relations (in the class of pro- ST-groups), where n -- k ~ 2; then Aut n G = InnG. Hence there follows, for instance, the above-mentioned coincidence for free profinite groups [3], for free prosoluble groups, and for free pro-p-groups for arbitrary p. In [2] the coincidence Aut n = InnG(k) was also proved for the absolute Galois group G(k) of a finite extension k of the field of p-adic numbers Qp. The question of the validity of the analog of Neukirch's theorem in this case remains open. Concerning normal automorphisms of discrete groups, only one fact is known: Lubotzky [4] and Lue [5] proved in different ways that Aut n F = I n n F if F is a free non-Abelian group. We note only one particularity: In a profinite group any normal subgroup (we are talking, naturally, only of closed subgroups) is an intersection of normal subgroups of finite indices; therefore, a normal automorphism leaves unchanged the whole lattice of normal subgroups. For discrete groups this is so only when the group G is capable of being approximated in a hereditary finite way, that is, when G can be finitely approximated together with all its homomorphic images. In the general case, a normal automorphism leaves invariant the lattice of normal subgroups of finite indices. Moreover, it ought to be borne in mind that the coincidence Aut n G = I n n G was established only for profinite groups having well-defined properties of universality: Their finite homomorphic images could be "built up from below" into irreducible modules over fields of nonzero characteristic, and thereupon the theory of finite group representations could be applied. In the discrete case it is necessary that the group G itself should have some well-defined approximation properties (say, the property of being approximable with respect to conjugacy) and, moreover, that it should contain sufficiently well organized normal subgroups closed in the topology defined by the normal subgroups of finite indices, being called by us a cofinite topology. Because of this, in the present paper we concentrate our attention on groups of the type F/R', and in the first place on free soluble groups. The main results are as follows: THEOREM
I.
If S is a free non-Abelian
soluble group, then Aut n S = Inn S.
THEOREM 2. If F/R is a finitely generated nilpotent noncyclic group without torsion, then Aut nF/R' = Inn F/R'. THEOREM 3. If F/R is a polycyclic noncyclic group without torsion, if R~__F ', and if the units of the group ring ZF/R are of the form • where g ~ F / R , then Aut nF/R' = InnF/R'. The coincidence Aut n G = I n n G for a group G means that an automorphism of the group G that induces an identity mapping of the lattice of normal subgroups of finite indices is an Complex Division of the Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR, Omsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 24, No. 4, pp. 138-149, July-August, 1983. Original article submitted June 6, 1981.
604
0037-4466/83/2404-0604507.50
9 1984 Plenum Publishing
Corporation
inner automorphism. In the present paper fairly general statements are obtained about the properties of normal automorphisms of groups of the type F/R'; they are exactly formulated in Sec. 3. We also considered strongly normal automorphisms inducing identity mappings on the lattice of all normal subgroups. We proved that under well-defined conditions on the group F/R strongly normal automorphisms of the group F/R' are inner automorphisms. 2.
Normal Automorphisms
of Free
Soluble
Groups
The aim of this section is proving Theorem I. However, we shall begin with some preliminaries: We shall elicidate the structure of groups of normal automorphisms of free Abelian and met-Abelian groups in order to secure a basis of induction for the proof of Theorem I, and we shall deduce the technical propositions that are indispensable for this proof. Proposition I. Let A be a free Abelian group. Then an automorphism inverting all the elements of the group A~
Aut~A ~ Z / 2 Z ~
gr(i),
where
i is
Proof. Let ~ A u t ~ A , and let a be an element of some basis of A. It suffices to prove that if ~ i d , then a~ = a -~ . Let A = gr (a) K A ~ , a ~:a~a~, ~ Z , ~A~. If a ~ 1 , we choose a subgroup N ~ A ~ having a finite index and not containing c . Then (gr (a) X N ) ~ gr (a) X N , which contradicts the normality of 9 . Thus, a ~ a k, and hence k ~ { • 9 Let cl and a2 be elements of some basis of A, and a ~ a l , a m 2 = a ~- I . Consider the subgroup B = gr (a1~Ti)-(gr(a~) • g r ( ~ ) ) • A12, where A = gr(~i) • gr(~2) x Ai2 with m > 2. Obviously, B ~B, which again leads to a contradiction. It follows that a
