MATHEMATICS

V. G. Vilyatser

STABLE GROUPS OF AUTOMORPHISMS

(Presented by Academician A. I. Mal'tsev on 1 XII 1959)

§ 1. In the work (¹) L. A. Kaluzhnin proved the following theorem:
If a group \(G\) has a finite invariant series stable with respect to its group of automorphisms \(\Phi\), then the group \(\Phi\) is nilpotent.

In the case when the \(\Phi\)-stable series is normal, only the solvability of the group \(\Phi\) has been proved; the question of its nilpotency remains open for the time being. In the present note we consider some cases in which the group \(\Phi\) turns out to be nilpotent or locally nilpotent. The main result is the following theorem:

An externally nilpotent group of automorphisms of a group with the maximality condition is nilpotent.

We give some definitions and notation (see also (²–⁴)). A set of automorphisms \(\Sigma\) of a group \(G\) is called stable if in \(G\) there is an ascending normal \(\Sigma\)-stable series. If in \(G\) there is a local system of \(\Sigma\)-admissible subgroups in which \(\Sigma\) induces stable sets of automorphisms, then \(\Sigma\) is called a locally stable set of automorphisms. If the \(\Sigma\)-stable series is finite, then \(\Sigma\) is called an externally nilpotent set. In particular, the set \(\Sigma\) may consist of only one automorphism, or may be some group of automorphisms of the group \(G\).

Let \(g\) and \(h\) be elements of the group \(G\); \(\varphi\) and \(\sigma\) elements of its group of automorphisms \(\Phi\). We use the usual notation for commutators; analogously,
\([g,\varphi(0)] = g\), \([g,\varphi(n)] = [[g,\varphi(n-1)],\varphi]\), and so on. \([G,\Phi]\) is the \(\Phi\)-commutant of the group \(G\), i.e., the subgroup of the group \(G\) generated by all commutators \([g,\varphi]\). By \(\hat g\) we denote the inner automorphism induced by the element \(g\) in the group \(G\). The identity of the group \(G\) is denoted by \(e\), that of the group \(\Phi\) by \(\varepsilon\).

§ 2. Lemma 1. Let \(G\) be a group; \(\varphi\) and \(\sigma\) automorphisms of this group; \(E \subset H \subset G\) a normal series admissible with respect to these automorphisms, with \(\sigma\) inducing identity automorphisms in the factors of this series, and \(\varphi\) in the factor \(G/H\). Let \(g \in G\), \([g,\varphi]=h\). Then the following relation holds:

\[ [g,[\sigma,\varphi(n)]]=[[g,\sigma]\varphi^{-1}\hat h(n)] \qquad (n=0,1,2,\ldots). \tag{*} \]

Proof. Denote \([\sigma,\varphi(n)] = \sigma_n\), \([g,\sigma_n] = a_n\) \((n=0,1,2,\ldots)\). We have:
\([g,\sigma_n]=g^{-1}\sigma_n(g)=a_n\), whence

\[ \sigma_n(g)=ga_n. \]

On the other hand,

\[ \begin{aligned} \sigma_n(g) &= [\sigma_{n-1},\varphi](g) = \sigma_{n-1}^{-1}\varphi^{-1}\sigma_{n-1}\varphi(g) = \sigma_{n-1}^{-1}\varphi^{-1}\sigma_{n-1}(gh) \\ &= \sigma_{n-1}^{-1}\varphi^{-1}(ga_{n-1}h) = \sigma_{n-1}^{-1}\varphi^{-1}(gh\cdot h^{-1}a_{n-1}h) \\ &= \sigma_{n-1}^{-1}\bigl(g\varphi^{-1}(h^{-1}a_{n-1}h)\bigr) \\ &= ga_{n-1}^{-1}\varphi^{-1}(h^{-1}a_{n-1}h) = ga_{n-1}^{-1}\varphi^{-1}\hat h(a_{n-1}). \end{aligned} \]

Comparing the expressions for \(\sigma_n(g)\), we obtain:

\[ a_n=a_{n-1}^{-1}\varphi^{-1}\hat h(a_{n-1}). \tag{**} \]

If \(n=0\), then \((*)\) is obviously true. Now from \((**)\) it is easy to obtain the validity of \((*)\) for any \(n\).

Lemma 2. Let \(H\) be a locally nilpotent group, \(\varphi\) its nil-automorphism, and \(h\in H\). Then \(\varphi\hat h\) is also a nil-automorphism of the group \(H\).

Proof. Let \(\bar H=\{H,\bar\varphi\}\) be the cyclic extension of the group \(H\) by means of the element \(\bar\varphi\), inducing in \(H\) the automorphism \(\varphi\). Obviously, \(\bar\varphi\) will be a nil-element of the group \(\bar H\), and therefore the group \(\bar H\) is locally nilpotent \((^5)\). But then the element \(h\bar\varphi\) will be a nil-element, and the inner automorphism \(\widehat{h\bar\varphi}\) a nil-automorphism of the group \(\bar H\). It remains to note that the automorphism \(\widehat{h\bar\varphi}\) in the group \(H\) induces the automorphism \(\varphi\hat h\).

Remark. If in the hypothesis of the lemma \(H\) is a nilpotent \(M\)-group (\(M\)-groups are groups with the maximality condition), then \(\bar H\) is also a nilpotent \(M\)-group, whence it follows that the automorphism \(\varphi\hat h\) has finite nilpotency index, i.e., there exists an \(n\geqslant 0\) such that for every \(a\in H\) the equality \([a,\varphi\hat h(n)]=e\) holds.

Lemma 3. Let \(H\) be a nilpotent group, and let \(\Phi\) be its outer nilpotent group of automorphisms. Then the group \(\Phi\) is nilpotent.

The proof of this lemma is not difficult to obtain by means of a theorem of L. A. Kaluzhnin.

Theorem 1. Let \(G\) be an \(M\)-group, and let \(\Phi\) be its outer nilpotent group of automorphisms. Then the group \(\Phi\) is nilpotent.

Proof. In \((^3)\), B. I. Plotkin proved that, under the hypotheses of the theorem, \([G,\Phi]\) is a nilpotent group. Denote by \(\Sigma\) the invariant subgroup of the group \(\Phi\) consisting of all automorphisms that induce the identity automorphisms in \([G,\Phi]\). The factor group \(\Phi/\Sigma\) is isomorphic to the group of automorphisms of the group \([G,\Phi]\) induced in it by the group \(\Phi\), and, by Lemma 3, is nilpotent. We shall show that \(\Phi\) is a nil-group. Let \(\varphi\) and \(\sigma\) be arbitrary automorphisms from \(\Phi\). Denote \([\sigma,\varphi(i)]=\sigma_i\). In view of the nilpotency of the group \(\Phi/\Sigma\), there exists a \(k\geqslant 0\) such that \(\sigma_k\in\Sigma\). Let \(g\in G\), \([g,\sigma_k]=a\), \([g,\varphi]=h\). By the remark to Lemma 2, there exists an \(n\geqslant 0\), independent of the choice of the element \(g\), such that \([a,\varphi^{-1}\hat h(n)]=e\), whence, by Lemma 1, we have
\[ [g,[\sigma^k,\varphi(n)]]=[g,[\sigma,\varphi(k+n)]]=e. \]
Since the last equality holds identically for all \(g\in G\), it follows that \([\sigma,\varphi(k+n)]=e\), i.e. \(\Phi\) is a nil-group. But an outer nilpotent group of automorphisms of an \(M\)-group is itself an \(M\)-group \((^6)\), and therefore the group \(\Phi\) is nilpotent \((^7)\).

Corollary. The group of nil-automorphisms of an \(M\)-group is nilpotent \((^3)\).

§ 3. In this section we consider groups of locally stable automorphisms. The following theorem clarifies the role of locally stable elements of a group, i.e., of those elements that induce inner locally stable automorphisms in the group.

Theorem 2. In an arbitrary group, the radical coincides with the set of all locally stable elements.

Proof. 1) We shall show that every element of the radical \(R(G)\) of the group \(G\) is locally stable. Let \(F\) be the subgroup generated by an arbitrary finite set of elements from \(G\) and by an element \(h\in R(G)\). The radical \(R(F)\) is a countable locally nilpotent group; therefore in \(R(F)\) one can construct an ascending normal series beginning with the cyclic group \(\langle h\rangle\) and reaching \(R(F)\). It follows that \(\langle h\rangle\) is subinvariant in \(F\), i.e. the element \(h\) induces a stable automorphism in \(F\).

2) We shall show the converse, i.e. that every locally stable element

\(h \in G\) belongs to the radical \(R(G)\). Let \([G^{(\alpha)}]\) be a local system of subgroups of the group \(G\), in which \(\hat h\) induces stable automorphisms,

\[ E = G_0^{(\alpha)} \subset G_1^{(\alpha)} \subset \ldots \subset G_\beta^{(\alpha)} \subset \ldots \subset G_\gamma^{(\alpha)} = G^{(\alpha)} \]

an \(\hat h\)-stable series of subgroups of \(G^{(\alpha)}\). Denote, for all \(\beta\),
\(\{G_\beta^{(\alpha)}, h\}=H_\beta^{(\alpha)}\). Then the cyclic subgroup \(\langle h\rangle\) will be subinvariant in \(H_\gamma^{(\alpha)}=H^{(\alpha)}\). But in this case \(\langle h\rangle \subset R(H^{(\alpha)})\), and, since \([H^{(\alpha)}]\) is a local system of the group \(G\), \(\langle h\rangle \subset R(G)\) \((^8)\).

Corollary. If \(G\) is an \(LM\)-radical group, then every one of its nil-automorphisms \(\varphi\) is locally stable.

Proof. Let \(\overline G=\{G,\overline\varphi\}\) be the cyclic extension of the group \(G\) by means of the element \(\overline\varphi\), inducing in the group \(G\) the automorphism \(\varphi\). In the \(LM\)-radical group \(\overline G\), the element \(\overline\varphi\) is a nil-element and therefore is contained in the radical \(R(\overline G)\) \((^9)\). But then, by Theorem 2, \(\overline\varphi\) is a locally stable element of the group \(\overline G\) and, evidently, induces in the group \(G\) a locally stable automorphism.

Lemma 4. Let \(\Phi\) be a periodic group of locally stable automorphisms of the group \(G\). Then \([G,\Phi]\) is a periodic locally nilpotent group.

Proof. It follows from \((^3)\) that, under the conditions of the lemma, \([G,\Phi]\) is a locally nilpotent group. Let \(P\) be the periodic part of \([G,\Phi]\) and \(\varphi(g)=gh\), where \(\varphi\in\Phi\), \(g\in G\), \(h\in [G,\Phi]\setminus P\). If \(\varphi^n=\varepsilon\), then \(\varphi^n(g)=g=gh^n a\), where \(a\in P\). But the equality \(h^n a=e\) is impossible; therefore \([G,\Phi]=P\).

Lemma 5. A finite group of nil-automorphisms of a periodic locally nilpotent group is nilpotent.

Theorem 3. A finite group of locally stable automorphisms \(\Phi\) of an arbitrary group \(G\) is nilpotent.

Proof. We shall show that \(\Phi\) is a nil-group. Denote by \(\Sigma\) the set of automorphisms inducing identical automorphisms in \([G,\Phi]\). Then \(\Phi/\Sigma\) is a nilpotent group (Lemmas 4 and 5). Let \(\varphi\in\Phi\), \(\sigma\in\Phi\), \([\sigma,\varphi(i)]=\sigma_i\). For some \(k\ge 0\) we have \(\sigma_k\in\Sigma\). Suppose that \(\sigma_i\ne\varepsilon\) for no \(i\). Then, in view of the finiteness of the group \(\Sigma\), there is an \(s>0\) such that \(\sigma_{m+s}=\sigma_m\) \((m\ge k)\), and, since \(\sigma_m\ne\varepsilon\), there is a \(g\in G\) such that \([g,\sigma_m]=a\ne e\), whence \([g,\sigma_{m+j}]\ne e\) for no \(j\ge 0\). Denote \([g,\varphi]=h\). By Lemma 2, \(\varphi^{-1}\hat h\) is a nil-automorphism; therefore for some \(n\ge 0\) we have \([a,\varphi^{-1}\hat h(n)]=e\). But then, by Lemma 1, \([g,[\sigma_m,\varphi(n)]]=[g,\sigma_{m+n}]=e\). The contradiction obtained proves the theorem, since a finite nil-group is nilpotent.

Corollary 1. A locally finite group of locally stable automorphisms is locally nilpotent.

Corollary 2. A locally finite group of nil-automorphisms of an \(LM\)-radical group is locally nilpotent.

Remark. Corollary 2 refines a result obtained earlier by the author \((^4)\).

Ural State University
named after A. M. Gorky

Received
23 X 1959

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