MATHEMATICS

B. I. Plotkin and V. G. Vilyatser

ON THE THEORY OF LOCALLY STABLE GROUPS OF AUTOMORPHISMS

(Presented by Academician A. I. Mal'cev, 9 V 1960)

1. As is known, there are deep connections between the external and internal properties of groups of automorphisms. The present paper is devoted to conditions under which an external property—the local stability of a group of automorphisms—entails its internal local nilpotency. This question has already been considered by a number of authors.

The notion of a locally stable group of automorphisms is the result of transferring the notion of a locally nilpotent group to the case of a pair ((G,\Phi)), where (\Phi) is a group of automorphisms of the group (G). We give the definitions used in the paper (see also ((^1))).

A set (group) of automorphisms (\Sigma) of a group (G) is called stable (relative to (G)) if in (G) there is an ascending normal (\Sigma)-admissible series in whose factors (\Sigma) induces the identity automorphisms (identities). (\Sigma) is called finitely stable if there is a corresponding finite series. A finite set (\Sigma) is called locally stable if in (G) there is a local system of (\Sigma)-admissible subgroups, in each of which (\Sigma) acts as a stable set. Finally, the group (\Phi) is called locally stable if every finite subset of (\Phi) is locally stable. Locally finitely stable groups of automorphisms are defined in the corresponding way.

1. Theorem 1*. A finitely stable group of automorphisms of a free group is nilpotent.

Proof. 1) We first note the following (known) fact. Let (G) be a group, (H) its normal divisor, and let (\sigma) be an automorphism of the group (G) inducing the identity on (H) and on (G/H). Then for any element (g \in G) the element
[
[g,\sigma] = g^{-1}\cdot \sigma(g)
]
is contained in the center of (H). Indeed, if (h \in H), then
[
\begin{aligned}
h^{-1}[g,\sigma]h
&= h^{-1}\cdot g^{-1}\cdot \sigma(g)\cdot h
= h^{-1}\cdot g^{-1}\cdot \sigma(gh) \
&= h^{-1}\cdot g^{-1}\cdot \sigma(ghg^{-1}\cdot g)
= h^{-1}\cdot g^{-1}\cdot ghg^{-1}\cdot \sigma(g)
= g^{-1}\cdot \sigma(g)
= [g,\sigma].
\end{aligned}
]

From the computations just given it is also easy to obtain the following assertion:

If in a group (G) there is an ascending invariant (\Phi)-stable series, then the commutant ([G\Phi]) belongs to the centralizer of this series and, in particular, itself possesses an ascending central series.

* 2) Let (G, H), and (\sigma) be as before, and suppose, in addition, that (\varphi) is an automorphism of the group (G) inducing the identity in (G/H). Then for any (g \in G) the relation
[
[g,[\sigma,\varphi]] = [[g,\sigma],\varphi^{-1}]
]
holds.

* After the paper had already been sent to press, we became aware of a paper by P. Hall ((^7)), in which a theorem coinciding with our Theorem 1 is proved. In this connection Theorem 1 should be regarded as a new proof of Hall’s theorem. Unlike Hall, we give a proof without passing to the holomorph.

For the proof, denote ([g,\sigma]=a,\ [g,\varphi]=h). We have

[
\begin{aligned}
[g,[\sigma,\varphi]]
&=g^{-1}\cdot\sigma^{-1}\varphi^{-1}\sigma\varphi(g)
=g^{-1}\cdot\sigma^{-1}\varphi^{-1}\sigma(gh) \
&=g^{-1}\cdot\sigma^{-1}\varphi^{-1}(g a\cdot h)
=g^{-1}\cdot\sigma^{-1}\varphi^{-1}(g h a) \
&=g^{-1}\cdot\sigma^{-1}(g\cdot\varphi^{-1}(a))
=g^{-1}\cdot g\cdot a^{-1}\cdot\varphi^{-1}(a)
=a^{-1}\cdot\varphi^{-1}(a)=[[g,\sigma],\varphi^{-1}],
\end{aligned}
]

as was required. The relation obtained improves the formula of the lemma from ((^{2})).

3) Let, further, (G) be a group, (H) its normal divisor, and (\Phi) a group of automorphisms of the group (G) such that (H) is (\Phi)-admissible and in (G/H) the group (\Phi) induces the identity. Let, moreover, (\Sigma) be a normal divisor in (\Phi) inducing the identity in (H). Then the equality holds (for (g\in G))

[
[g,[\Sigma,\Phi]]=[[g,\Sigma],\Phi].
]

Indeed, since ([\Sigma,\Phi]\subset \Sigma), and (\Sigma) induces the identity in (G/H) and in (H), every element of the left-hand side has the form ([g,\psi]), where (\psi) is an arbitrary element of ([\Sigma,\Phi]). The element (\psi) has the form

[
\psi=\prod_i[\sigma_{k_i},\varphi_{j_i}]^{\varepsilon_i},
]

where (\sigma_{k_i}\in\Sigma,\ \varphi_{j_i}\in\Phi), and (\varepsilon_i=\pm 1). We have

[
[g,\psi]=\left[g,\prod_i[\sigma_{k_i},\varphi_{j_i}]^{\varepsilon_i}\right]
=\prod_i[g,[\sigma_{k_i},\varphi_{j_i}]]^{\varepsilon_i}
=\prod_i [[g,\sigma_{k_i}],\varphi_{j_i}^{-1}]^{\varepsilon_i}\in [[g,\Sigma],\Phi],
]

which proves the inclusion in one direction. The second inclusion is checked analogously.

Let us now put (\Sigma_i=[\Sigma,\Phi(i)]). The subgroup (\Sigma_i) is a normal divisor in (\Phi) and, just like (\Sigma), induces the identity in (H) and (G/H). Consequently,

[
[g,[\Sigma_i,\Phi]]=[[g,\Sigma_i],\Phi].
]

Hence, by induction, we obtain the relation

[
[g,[\Sigma,\Phi(n)]]=[[g,\Sigma],\Phi(n)]\qquad (n=0,1,2,\ldots).
]

4) We pass to the proof of the theorem. Let

[
E=G_0\subset G_1\subset\cdots\subset G_n=G
]

be a finite normal (\Phi)-stable series in the group (G). We shall prove the nilpotency of the group (\Phi) by induction on the length of such a series. For (n=1,2) the assertion is obvious. Suppose that it is true for (k\le n-1). Let (\Sigma) be the (\Phi)-centralizer of the subgroup (G_{n-1}). Then (\Sigma) is abelian and (\Phi/\Sigma), by the induction hypothesis, is nilpotent. From the preceding it follows that ([\Sigma,\Phi(n-1)]) is the identity in (\Phi), i.e. (\Phi) is a nilpotent group.

Recall that in the work of L. A. Kaluzhnin ((^{4})) an analogous theorem was proved for the case of an invariant stable series.

3. Theorem 2. Let (\Phi) be a locally stable group of automorphisms of a group (G). Suppose that in the radical of the group (G) the periodic part is finite. Then the group (\Phi) is locally nilpotent if and only if it has a local system consisting of subgroups of finite rank.

The work ((^{5})) contains an example showing that local nilpotency of the group (\Phi) cannot be derived from local stability of this group alone. On the other hand, it is obvious that every locally nilpotent group possesses a local system consisting of subgroups of finite rank.

We give auxiliary propositions.

Lemma 1. Let (\Phi) be a locally stable group of automorphisms of a group (G). Let the periodic part (P) of the radical of the group (G) be finite and have order (m), and let (\Phi) be a group of finite rank (r). Then the set of all elements of finite order in (\Phi) is a subgroup of (\Phi), coinciding with the (\Phi)-centralizer of the factor group (G/P). This subgroup is finite and has order not exceeding the number (m!\,m^r).

The proof of this lemma is obtained from the following considerations. In ((^5)) it is shown that every element of finite order from a locally stable group of automorphisms of some group (G) induces the identity automorphism in the factor group (G/P(R(G))), where (P(R(G))) is the periodic part in (R(G)). Denoting by (\Psi) the (\Phi)-centralizer of the factor group (G/P), we see that all elements of finite order from (\Phi) belong to (\Psi). The order of (\Psi) is estimated as follows.

Let (\Sigma) be the (\Psi)-centralizer of the subgroup (P). Then the order of (\Psi/\Sigma) does not exceed (m!). It is easy to see that (\Sigma) is a periodic group, and the orders of all elements of (\Sigma) are bounded by the number (m). Since (\Sigma) is an abelian group of rank not exceeding (r), it is clear that the order of (\Sigma) does not exceed (m^r).

Lemma 2. Let (\Phi) be a stable group of automorphisms of the group (G). Then, if (\Phi) has a finite number of generators and finite rank, and in (G) the periodic part of the radical is finite, then (\Phi) is nilpotent.

Proof. Let (r) be the rank of (\Phi), (m) the order of the periodic part of the radical (R(G)), and

[
E=G_0 \subset G_1 \subset \cdots \subset G_\alpha \subset G_{\alpha+1} \subset \cdots \subset G_\gamma=G
]

a (\Phi)-stable series of the group (G). From Lemma 2 it follows that, both in the group (\Phi) itself and in every group of automorphisms induced by the group (\Phi) in the groups (G_\alpha), the periodic parts are finite and their orders do not exceed (m!\,m^r). We shall prove the nilpotency of the group (\Phi) by induction on the length of the (\Phi)-stable series. For (\gamma=1) the assertion is obvious; suppose that it is true for all (\alpha<\gamma). Let there exist (\beta=\gamma-1). Denote by (\mathfrak Z) the (\Phi)-centralizer of (G_\beta). Then (\Phi/\mathfrak Z) is nilpotent by the induction hypothesis, and (\mathfrak Z) is abelian. We shall show that every element (\varphi\in\Phi) induces a nil-automorphism in (\mathfrak Z). Let (\sigma) be an arbitrary element in (\mathfrak Z), and let (g) be some fixed element in (G). The mapping (\sigma\to[g,\sigma]) is a homomorphism of the group (\mathfrak Z) onto a central subgroup (A) of (G_\beta). (A) has finite rank, not exceeding (r), and since, according to ((^6)), (A) lies in the radical of the group (G), the periodic part of (A) has order not exceeding (m). From the relation

[
[[g,\sigma],\varphi^{-1}]=[g,[\sigma,\varphi]]
]

it follows that (A) is (\Phi)-admissible. It is now clear that, if (s=r+m), then ([A,\Phi(s)]=E), and we emphasize that (s) does not depend on (A), i.e. on the choice of the element (g). Thus, from item 3) of the proof of Theorem 1 it follows that the automorphism ([\sigma,\varphi(s)]) acts identically on every element (g\in G). Hence, by the finiteness of the number of generators, it follows that (\Phi) is a nilpotent group ((^3)).

It remains to consider the case when (\gamma) is a limit ordinal. For this it is enough to refer to the following proposition.

Lemma 3. Let the group (G) have a local system of (\Phi)-admissible subgroups (G_\alpha), in each of which (\Phi) induces a nilpotent group of rank not exceeding (r), whose periodic part is finite and whose order does not exceed the number (k). Then the whole group (\Phi) is nilpotent.

Proof. Let (\mathfrak Z_\alpha) be the (\Phi)-centralizer of the subgroup (G_\alpha). The intersection of all (\mathfrak Z_\alpha) coincides with the identity of the group (\Phi); therefore (\Phi) is a subdirect product of the groups (\Gamma_\alpha \simeq \Phi/\mathfrak Z_\alpha). All (\Gamma_\alpha) have nilpotency class not exceeding (r+k). But a subdirect product of nilpotent groups whose nilpotency classes are bounded in the aggregate is again a nilpotent group. Thus (\Phi) is a nilpotent group. In particular, Lemma 2 is also proved.

We pass to the proof of the theorem. Let (G) and (\Phi) satisfy the hypotheses of the theorem. Without loss of generality one may assume that (\Phi) has a finite number of generators and, consequently, finite rank.

Let ({G_\alpha}) be a local system of (\Phi)-admissible subgroups, in each of which (\Phi) acts as a stable group. All commutants ([G_\alpha,\Phi]) belong to the radical of the group (G) and, consequently, have periodic parts whose orders do not exceed the order of the periodic part of the radical. Let (\mathfrak Z_\alpha) be the (\Phi)-centralizer of (G_\alpha). Since in Lemmas 1 and 2 the radical can be replaced by the commutant ([G,\Phi]), it now follows from these lemmas that all the groups (\Phi/\mathfrak Z_\alpha) are nilpotent and that the orders of their periodic parts are bounded by some number (k). Since the group (\Phi) itself has finite rank, by Lemma 3 it is nilpotent.

REFERENCES

(^{1}) B. I. Plotkin, UMN, 13, 4, 82 (1958).
(^{2}) V. G. Vilyatser, DAN, 131, No. 4 (1960).
(^{3}) B. I. Plotkin, UMN, 9, 3, 181 (1954).
(^{4}) L. Kalujnin, Ber. Math. Tagung, Berlin, 164 (1953).
(^{5}) V. G. Vilyatser, Uchen. zap. Ural. gos. univ. im. A. M. Gor’kogo, issue 23 (1959).
(^{6}) B. I. Plotkin, DAN, 130, No. 5 (1960).
(^{7}) F. Holl, Illinois J. Math., 2, No. 4 B (1958).