B. I. Plotkin

On Infinite-Dimensional Linear Groups

(Presented by Academician A. I. Mal’cev, 3 VI 1963)

1. A linear group is a subgroup of the group of invertible linear operators (automorphisms) of a vector space. If the space is finite-dimensional, then the corresponding group is called finite-dimensional, and in the case of an infinite-dimensional space the group is called infinite-dimensional. The following general problem naturally arises: to study infinite-dimensional groups which, in one sense or another, are close to finite-dimensional ones. Some classes of such groups are considered in the present note.

The following notation and definitions are used in the note: \(G\) is a vector space over a certain field \(P\); \(\Gamma\) is a certain group of automorphisms of this space; \([\Gamma]\) is the linear envelope of \(\Gamma\) in the algebra of all linear operators of the space \(G\); \(\Sigma\) is a certain subgroup of \(\Gamma\) having a finite system of generators.

A system of \(\Gamma\)-admissible subspaces of the space \(G\)

\[ G_0 = 0 \subset \ldots \subset G_\alpha \subset G_{\alpha+1} \subset \ldots \subset G_\gamma = G \]

is called stable relative to \(\Gamma\) if, for every \(\sigma \in \Gamma\) and every jump of the system, one has
\[ G_{\alpha+1}\circ(\sigma-\varepsilon)\subset G_\alpha; \]
\(\varepsilon\) is the identity in \(\Gamma\).

The group \(\Gamma\) is called weakly stable (or stable, or externally nilpotent) if in \(G\) there is a system of subspaces stable relative to \(\Gamma\) (respectively, either an ascending stable series, or a finite stable series). \(\Gamma\) is called locally stable if, for every \(\Sigma\) in \(G\), there is a local system of \(\Sigma\)-admissible subspaces \(H_\alpha\) such that in each \(H_\alpha\), \(\Sigma\) acts as a stable group. It can be shown that if \(\Gamma\) is locally stable, then all such \(H_\alpha\) may be taken to be finite-dimensional. The group \(\Gamma\) is called locally externally nilpotent if every \(\Sigma\) is nilpotent relative to \(G\). A pair of \(\Gamma\)-admissible subspaces \(A\) and \(B\), \(A\subset B\), of \(G\) is called a compositional pair if there are no other \(\Gamma\)-admissible subspaces between them. We shall denote by \(\alpha(\Gamma)\) the set of all elements \(\sigma\) of \(\Gamma\) such that, for every compositional pair \(A,B\), \(A\subset B\), one has
\[ B\circ(\sigma-\varepsilon)\subset A. \]
\(\alpha(\Gamma)\) is a normal divisor in \(\Gamma\), called the external radical of this group. By \(\beta(\Gamma)\) is denoted the locally stable radical of the group \(\Gamma\)—the subgroup in \(\Gamma\) generated by all its locally stable normal divisors. \(\gamma(\Gamma)\) is the subgroup in \(\Gamma\) generated by all locally externally nilpotent normal divisors; this subgroup is itself locally externally nilpotent. Some general relations among the three indicated radicals were considered in the note \((^1)\).

We now define finiteness conditions. We shall call the group \(\Gamma\) \(c\)-finite if in \(G\) there is a \(\Gamma\)-admissible finite-dimensional subspace \(H\) such that the induced representation of \(\Gamma\) relative to \(H\) is faithful; we shall call \(\Gamma\) \(b\)-finite if in \(G\) there is a local system of \(\Gamma\)-admissible finite-dimensional subspaces; we shall call \(\Gamma\) \(a\)-co-

finite, if \([\Gamma]\) is a finite-dimensional algebra. It is easy to see that \(a\)-finiteness is here the strongest finiteness condition. The notions of local \(c\)-finiteness, local \(b\)-finiteness, and local \(a\)-finiteness of the group \(\Gamma\) are now defined in the corresponding way.

1. From the general results of note \((^1)\) it follows that if \(\Gamma\) is a locally \(b\)-finite group, then \(\alpha(\Gamma)=\beta(\Gamma)\), so that \(\alpha(\Gamma)\) has a central system and is a locally stable group \((^1)\). We shall show that if \(\Gamma\) is locally \(c\)-finite, then \(\alpha(\Gamma)\) is a locally nilpotent group, while \(\beta(\Gamma)\) is a locally stable group and belongs to \(\alpha(\Gamma)\).

Let \(\Phi\) be a locally \(c\)-finite weakly stable group of automorphisms of the space \(G\), and let \(\Sigma\) be a subgroup of \(\Phi\) having a finite number of generators. Find in \(G\) a finite-dimensional \(\Sigma\)-admissible subspace \(H\) such that the pair \((H,\Sigma)\) is exact. Since \(\Sigma\) is a weakly stable group and since \(H\) is finite-dimensional, \(\Sigma\) is nilpotent relative to \(H\). In view of the fact that the pair \((H,\Sigma)\) is exact, we conclude that \(\Sigma\) is nilpotent as an abstract group; hence \(\Phi\) is a locally nilpotent group.

Taking into account that \(\alpha(\Gamma)\) is a weakly stable group, in the case under consideration we obtain that \(\alpha(\Gamma)\) is a locally nilpotent group. The same can be said about \(\beta(\Gamma)\), since a locally stable group is always weakly stable. Let further \(R(\Gamma)\) be the locally nilpotent radical of the group \(\Gamma\). Since the radicals \(\alpha(\Gamma)\) and \(\beta(\Gamma)\), by what has been proved, belong to \(R(\Gamma)\), we obtain
\[ \alpha(\Gamma)=\alpha(R(\Gamma)) \quad \text{and} \quad \beta(\Gamma)=\beta(R(\Gamma)). \]
Keeping in mind, moreover, that a locally nilpotent group is a locally Noetherian group, on the basis of \((^1)\) we now conclude that \(\beta(\Gamma)\) is a locally stable group belonging to \(\alpha(\Gamma)\).

From known results concerning finite-dimensional linear groups it is easy to conclude that if \(\Gamma\) is a locally \(c\)-finite group, then the locally nilpotent radical \(R(\Gamma)\) coincides with the set of all nil-elements of the group \(\Gamma\). In addition, \(\Gamma\) has a locally solvable radical—the locally solvable normal divisor containing all locally solvable normal divisors of \(\Gamma\). It is easy to verify that this radical is a radical in the sense of A. G. Kurosh in the class of locally \(c\)-finite linear groups \((^2)\).

1. In the case when the group \(\Gamma\) is locally \(a\)-finite, all three radicals \(\alpha(\Gamma)\), \(\beta(\Gamma)\), and \(\gamma(\Gamma)\) coincide. The coincidence \(\alpha(\Gamma)=\beta(\Gamma)\) follows from the preceding remarks, and the equality \(\beta(\Gamma)=\gamma(\Gamma)\) is contained in the following proposition.

A locally stable group \(\Gamma\) is locally \(a\)-finite if and only if it is locally externally nilpotent.

Indeed, let \(\Gamma\) be a locally stable group, and suppose first that it is locally \(a\)-finite. Let \(\Sigma\) be a subgroup of \(\Gamma\) with a finite number of generators, and let \(n\) be the dimension of the algebra \([\Sigma]\). Take in \(G\) an arbitrary element \(g\), and let \(H\) be the subspace in \(G\) consisting of all elements of the form \(g\circ\varphi\), \(\varphi\in[\Sigma]\). Then \(H\) is a \(\Sigma\)-admissible subspace, and its dimension does not exceed \(n\). Since \(\Gamma\) is a locally stable group, \(\Sigma\) is nilpotent relative to \(H\). The length of a stable series in \(H\) cannot be greater than \(n\). This proves that for any \(g\in G\) all higher commutators of the form \([g;\sigma_1,\sigma_2,\ldots,\sigma_m]\) over all \(\sigma_i\in\Sigma\) with length \(m\geqslant n\) are transformed into zero in the space \(G\). Hence it follows that in \(G\) there is a stable series relative to \(\Sigma\) of length \(\leqslant n\). Thus \(\Gamma\) is a locally externally nilpotent group.

We prove the converse. Let \(\Gamma\) be locally externally nilpotent (by this, of course, it is said that \(\Gamma\) is locally stable). Take in \(\Gamma\) an arbitrary subgroup \(\Sigma\) with a finite number of generators \(\sigma_1,\sigma_2,\ldots,\sigma_k\). Since \(\Sigma\) is externally nilpotent, there exists an \(n\) such that the product of any \(n\) elements of the form \(\sigma_i-\varepsilon\) is equal to zero, so that the product of any \(n\) elements of the form \(\sigma_i,\ i=1,2,\ldots,k\), is expressed linearly in terms of products of such

elements with a smaller number of factors. Hence it obviously follows that the subalgebra \([\Sigma]\) has finite dimension.

Starting from the equality \(\alpha(\Gamma)=\gamma(\Gamma)\), analogously to how this is done in \((1)\), one can obtain the following further result:

If the group \(\Gamma\) is locally \(a\)-finite, then the outer radical of this group is connected with the radical of the Lie algebra \([\Gamma]\) by the formula

\[ \alpha(\Gamma)=\Gamma\cap (L([\Gamma])+\varepsilon). \]

Let us prove the following assertion.

In an arbitrary linear group \(\Gamma\), the product of an invariant locally \(a\)-finite subgroup \(\Phi_1\) and a locally \(a\)-finite subgroup \(\Phi_2\) is a locally \(a\)-finite subgroup.

First consider the case where \(\Phi_1\) and \(\Phi_2\) are \(a\)-finite. In this case, in \(\Phi_1\) and \(\Phi_2\) there are finite subsets, respectively \(X\) and \(Y\), such that \(X\) linearly generates \([\Phi_1]\) and \(Y\) linearly generates \([\Phi_2]\). We may also assume that \(X\) and \(Y\) contain the identity of the group \(\Gamma\). In this case the product of sets \(XY\) contains both \(X\) and \(Y\). Denote by \(S\) the linear hull of the set \(XY\). This hull is finite-dimensional and contains \([\Phi_1]\) and \([\Phi_2]\). We show that \(S\) is closed with respect to multiplication. Each element of \(\Phi_1\Phi_2\) has the form \(\varphi_1\varphi_2\), \(\varphi_1\in\Phi_1\), \(\varphi_2\in\Phi_2\). But \(\varphi_1\) and \(\varphi_2\) are expressed linearly in terms of elements of \(X\) and, respectively, \(Y\), so that \(\varphi_1\varphi_2\) is expressed linearly in terms of elements of \(XY\). Consequently, the subgroup \(\Phi_1\Phi_2\) belongs to \(S\). Hence follows the closedness of \(S\) with respect to multiplication, as well as the fact that \(\Phi_1\Phi_2\) is an \(a\)-finite group.

We pass to the general case. Let \(\Phi_1\) and \(\Phi_2\) be locally \(a\)-finite, and let \(\Sigma_2\) be a subgroup with a finite number of generators in \(\Phi_2\). Since this subgroup is \(a\)-finite, in it there is a finite subset \(Y\) linearly generating the subalgebra \([\Sigma_2]\). Next, let \(X\) be an arbitrary finite subset in \(\Phi_1\). Denote by \(Z\) the set of all elements of the form \(\sigma_2^{-1}\sigma\sigma_2\), \(\sigma\in X\), \(\sigma_2\in\Sigma_2\). Since \(Z\) is contained in the linear hull of the finite set \(YXY\), there is in \(Z\) a finite subset \(Z_0\) such that the linear hull of \(Z_0\) coincides with the linear hull of \(Z\). We may assume that \(X\subset Z_0\). Let \(\Sigma_1\) be the subgroup in \(\Phi_1\) generated by the set \(Z\), and let \(\Sigma_0\) be generated by \(Z_0\). It is clear that \([\Sigma_0]=[\Sigma_1]\), and, consequently, \(\Sigma_1\) is an \(a\)-finite group. Since the subgroup \(\Sigma_1\) is invariant with respect to \(\Sigma_2\), the group \(\Sigma_1\Sigma_2\) is \(a\)-finite. It is obvious that every subgroup with a finite number of generators from \(\Phi_1\Phi_2\) is contained in some such \(\Sigma_1\Sigma_2\), and the assertion is thereby proved.

From it follows the following theorem.

In an arbitrary linear group \(\Gamma\), the intersection of all maximal locally \(a\)-finite subgroups is an invariant locally \(a\)-finite subgroup in \(\Gamma\), containing all other such invariant subgroups.

This subgroup is naturally called the locally \(a\)-finite radical.

1. We now consider some properties of a locally nilpotent linear group. According to \((3)\), in such a group \(\Gamma\) the radical \(\beta(\Gamma)\) coincides with the set of all nilautomorphisms of the space \(G\) lying in \(\Gamma\). It is also easy to show that \(\gamma(\Gamma)\) is the totality of all \(\sigma\in\Gamma\) for which the endomorphism \(\sigma-\varepsilon\) is a nilpotent endomorphism.

We shall next prove the following proposition.

Let \(\Gamma\) be a locally nilpotent group of automorphisms of the vector space \(G\), and let \(\Phi\) be its invariant \(a\)-finite subgroup. Then, if \(\Phi\) is weakly stable, this subgroup has a finite central series of subgroups relative to the whole group \(\Gamma\).

For the proof we first note that from the weak stability and \(a\)-finiteness of \(\Phi\) follows the outer nilpotence of this subgroup. Let the series

\[ G=G^0\supset G^1\supset \ldots \supset G^n=0 \]

is a decreasing series of \(\Phi\)-commutants of the space \(G\). We shall prove the proposition by induction on the length of such a series. For length equal to 1 the assertion is obvious, and suppose that it has been proved for lengths \(< n\). From this assumption it is easy to derive that if \(\Sigma\) is the \(\Phi\)-centralizer of the subgroup \(G^1\), then in \(\Phi/\Sigma\) there is a finite central series in \(\Gamma/\Sigma\). It remains to show that in \(\Sigma\) there is also a finite central series relative to \(\Gamma\).

Denote by \(\Sigma^*\) the linear span of the set of all elements of the form \(\sigma-\varepsilon\), \(\sigma\in\Sigma\), and consider the mapping \(\sigma\to\sigma^*=\sigma-\varepsilon\in\Sigma^*\). A simple verification shows that such a mapping is an isomorphic embedding of the group \(\Sigma\) into the vector space \(\Sigma^*\). Denote by \(\Sigma'\) the image of \(\Sigma\) under this mapping in \(\Sigma^*\). Next define a representation of the group \(\Gamma\) by automorphisms of the vector space \(\Sigma^*\), putting \(\eta\circ\gamma=\gamma^{-1}\eta\gamma\) for \(\eta\in\Sigma^*\) and \(\gamma\in\Gamma\). It is easy to see that then the subgroup \(\Sigma'\subset\Sigma^*\) is \(\Gamma\)-admissible and that the mapping\(^*\) defines an isomorphism of the group pair \((\Sigma',\Gamma)\) and the internal pair \((\Sigma,\Gamma)\). Since the group \(\Gamma\) is locally nilpotent, the group pair \((\Sigma,\Gamma)\) is locally stable. Consequently, the pair \((\Sigma',\Gamma)\), and along with it also the pair \((\Sigma^*,\Gamma)\), is locally stable. Taking into account that the vector space \(\Sigma^*\) is finite-dimensional, by a known theorem we now conclude that \(\Gamma\) is nilpotent relative to \(\Sigma^*\). But then \(\Gamma\) is nilpotent also relative to \(\Sigma\), and this means that the required property of the subgroup \(\Sigma\) holds.

Let us note two consequences of this proposition.

By analogy with the abstract theory, we shall call a linear group \(\Gamma\) locally \(a\)-normal if in this group there is a local system of \(a\)-finite normal divisors.

If the group \(\Gamma\) is weakly stable and locally \(a\)-normal, then it has an increasing central series of length not greater than \(\omega\).

Another consequence is the following generalization of a theorem of Zassenhaus (\(^{4,5}\)).

Let \(\Gamma\) be a locally nilpotent group of automorphisms of a vector space \(G\), having an \(a\)-finite outer radical. Then, if in \(G\) there is a finite series of \(\Gamma\)-admissible subspaces, on all factors of which \(\Gamma\) acts as a nilpotent group, then the group \(\Gamma\) itself is nilpotent.

1. In all the cases considered above, the outer radical \(a(\Gamma)\) possesses a central system (being a \(Z\)-group). We shall show that in the general case this radical need not necessarily be a \(Z\)-group.

Let \(G\) be a countable-dimensional vector space over the field of rational numbers with basis \(e_1,e_2,\ldots,e_n,\ldots;\ e_0\), and let \(\sigma,\varphi,\psi_1,\psi_2,\ldots,\psi_n,\ldots\) be linear operators of this space defined by the formulas:

\[ \begin{gathered} e_{2k}\circ\sigma=e_{2k}+\frac{1}{2^k}e_0;\qquad e_{2k-1}\circ\sigma=e_{2k-1}+\frac{1}{2^{k-1}}e_0;\qquad e_0\circ\sigma=e_0;\\ e_{2k-1}\circ\varphi=e_{2k-1}+e_{2k}+e_{2k+1};\qquad e_{2k}\circ\varphi=e_{2k}+e_{2k+1};\qquad e_0\circ\varphi=e_0; \end{gathered} \quad \left\} k\geqslant 1;\right. \]

\[ e_i\circ\psi_j=e_i+e_{i+1}\ \text{for } 1\leqslant i<j;\qquad e_i\circ\psi_j=e_i\ \text{for } i\geqslant j \text{ and } i=0. \]

It is not difficult to verify that all these operators are automorphisms of the space \(G\). Denote by \(\Gamma\) the group generated by them. One can show that \(\Gamma=a(\Gamma)\). On the other hand, from the easily verified relation \(\varphi\sigma\varphi^{-1}\sigma^{-1}=\sigma\) it follows that the group \(\Gamma\) is not a \(Z\)-group.

Received
25 V 1963

REFERENCES

1. B. I. Plotkin, DAN, 144, No. 1, 52 (1962).
2. A. G. Kurosh, Sibirsk. matem. zhurn., 3, No. 6, 912 (1962).
3. B. I. Plotkin, DAN, 130, No. 5, 977 (1960).
4. H. Zassenhaus, Abh. Math. Sem. Hans. Univ. 12, No. 3—4, 312 (1938).
5. L. A. Suprunenko, R. I. Tyshkevich, Izv. AN SSSR, ser. matem., 24, 787 (1960).