MATHEMATICS

B. I. PLOTKIN

RADICALS ASSOCIATED WITH REPRESENTATIONS OF GROUPS

(Presented by Academician A. I. Mal’tsev, 23 XII 1961)

Let a certain representation \(f\) of a group \(\Gamma\) by automorphisms of a group \(\mathfrak G\) be given. We define the radical of this representation in the group \(\Gamma\). A pair of subgroups \(H\) and \(F\) (\(H \subset F\)) of \(\mathfrak G\) will be called a \(\Gamma\)-composition pair if in the group \(\mathfrak G\) there is a \(\Gamma\)-composition normal system in which the subgroups \(H\) and \(F\) constitute a jump. In this case the factor group \(F/H\) will be called a \(\Gamma\)-composition factor. It is not hard to see that, if the representation of \(\Gamma\) contains all inner automorphisms of the group \(\mathfrak G\), then the pair \(H, F\) will be a \(\Gamma\)-composition pair if and only if \(H\) and \(F\) are \(\Gamma\)-admissible normal divisors of the group \(\mathfrak G\) and the factor group \(F/H\) contains no proper \(\Gamma\)-admissible normal divisors of the group \(\mathfrak G/H\). In every \(\Gamma\)-composition factor \(F/H\) the representation \(f\) induces a certain irreducible representation of the group \(\Gamma\). Denote by \(\mathfrak Z(F/H)\) the kernel of such a representation, and by \(a(\Gamma,\mathfrak G)\) we shall denote the intersection of all \(\mathfrak Z_\Gamma(F/H)\) over all \(\Gamma\)-composition pairs of the group \(\mathfrak G\). The subgroup \(a(\Gamma,\mathfrak G)\) will be called the radical of the representation. It is clear that the factor group \(\Gamma/a(\Gamma,\mathfrak G)\) is a subdirect product of (irreducible) groups of automorphisms induced by the group \(\Gamma\) in the various \(\Gamma\)-composition factors of the group \(\mathfrak G\). An important problem is the study of the internal properties of the radical. Our immediate aim will be to consider the relation between the radical \(a(\Gamma,\mathfrak G)\) and the locally stable radical of the group \(\Gamma\) under certain restrictions. In the general case, as is not hard to show, these radicals are not incident.

Recall the definition of the locally stable radical \((^1)\). A group \(\Gamma\) is called stable if in \(\mathfrak G\) there is an ascending normal series of \(\Gamma\)-admissible subgroups, in all factors of which the elements of \(\Gamma\) induce the identity automorphisms. \(\Gamma\) is locally stable if, for every subgroup \(\Sigma\) of \(\Gamma\) with a finite number of generators in \(\mathfrak G\), there is a local system of \(\Sigma\)-admissible subgroups in which \(\Sigma\) acts as a stable group. We note here that, in the case when \(\mathfrak G\) is a soluble group, the stated property of the group \(\Sigma\) is equivalent to the stability of \(\Sigma\) with respect to the whole group \(\mathfrak G\) \((^1)\). The locally stable radical of the group \(\Gamma\) is the normal divisor in \(\Gamma\) generated by all locally stable normal divisors of \(\Gamma\). For the locally stable radical we shall use here the notation \(\beta(\Gamma,\mathfrak G)\).

Let \(\Phi\) be a subgroup in \(\Gamma\). We shall say that for the sequence \(\mathfrak G, \Gamma, \Phi\) condition (*) is satisfied if in the group \(\mathfrak G\) there is a normal system of \(\Gamma\)-admissible subgroups, in all factors of which the elements of \(\Phi\) induce the identity automorphisms. If, moreover, \(\Phi\) coincides with \(\Gamma\), then \(\Gamma\) is called a weakly stable group. We record the simplest cases in which condition (*) is satisfied.

1. If the locally nilpotent radical of the group \(\mathfrak G\) has finite special rank, then condition (*) is satisfied for the sequence \(\mathfrak G, \Gamma, \beta(\Gamma,\mathfrak G)\).

2. If \(\Phi\) is an externally nilpotent normal divisor in \(\Gamma\), then condition (*) is satisfied for the sequence \(\mathfrak G, \Gamma, \Phi\) \((^1)\).

Recall that \(\Phi\) is called an externally nilpotent group if in \(\mathfrak G\) there is a finite series stable with respect to \(\Phi\). Using the well-known method of A. I. Mal'cev \((^{2})\), it is not hard to show that condition \((*)\) for a fixed sequence \(\mathfrak G,\Gamma,\Phi\) can be written in the language of the narrow predicate calculus. As the underlying set \(M=(a,b,c,\ldots)\) one takes the set playing the role of indices of the terms of a suitable normal system. In this set two individual elements \(a_0\) and \(a_1\) must be fixed. The basic predicates defined on the set \(M\) are the order relation and all possible predicates of the form \(A_g(a)\), where the indices \(g\) run through the whole group \(\mathfrak G\), meaning, in content, that the element \(g\) belongs to the subgroup with index \(a\). The first group of axioms consists of axioms asserting that \(M\) is an ordered set. Then come axioms ensuring normality of the supplemented system. To these two groups of axioms one must add the following groups of axioms:

\[ (a)\,(A_g(a)\to A_{g\sigma}(a)),\qquad (Ea)\,(\bar A_g(a)\ \&\ \bar A_{g^{-1}\cdot g\sigma}(a)). \]

Here \(g\) ranges over the whole group \(\mathfrak G\), while \(\sigma\) in the first case ranges over the whole group \(\Gamma\), and in the second case over the subgroup \(\Phi\).

The existence of a model of such a system of axioms means that the required normal system exists in the group \(\mathfrak G\).

Relying on the indicated possibility of characterizing condition \((*)\), one easily obtains the following proposition.

Theorem 1. Let a representation \(f\) of the group \(\Gamma\) by automorphisms of the group \(\mathfrak G\) be given, and let \(\Phi\) be some subgroup in \(\Gamma\). Let also \(H\) and \(\Sigma\) be arbitrary subgroups with a finite number of generators in \(\mathfrak G\) and \(\Gamma\), respectively. Then, if all sequences of the form \(H^\Sigma,\Sigma,\Sigma\cap\Phi\) satisfy condition \((*)\), then the sequence \(\mathfrak G,\Gamma,\Phi\) also satisfies this same condition.

Here, as usual, \(H^\Sigma\) denotes the \(\Sigma\)-closure of the subgroup \(H\), and the representation of \(\Sigma\) with respect to \(H^\Sigma\) induced by the representation \(f\) is considered.

Theorem 2. The inclusion

\[ \beta(\Gamma,\mathfrak G)\subset \alpha(\Gamma,\mathfrak G) \]

holds in the following cases:

1) for every \(\Gamma\)-composition pair \(H,F\) of the group \(\mathfrak G\), the locally nilpotent radical has finite rank in the factor group \(F/H\);
2) all \(\Gamma\)-composition factors \(F/H\) are Abelian and \(\Gamma\) is an \(LM\)-group;
3) the representation \(f\) is algebraic.

The proof of the theorem is not difficult to obtain from the preceding considerations. We note only that a representation is called algebraic if, for every element \(g\in\mathfrak G\) and every finitely generated subgroup \(\Sigma\) of \(\Gamma\), the subgroup \(g^\Sigma\) has a finite number of generators. The representation of any locally finite group is algebraic. Hence, in particular, it follows that if the group \(\Gamma\) is locally finite, then \(\beta(\Gamma,\mathfrak G)\subset \alpha(\Gamma,\mathfrak G)\).

Theorem 3. If the groups \(\mathfrak G\) and \(\Gamma\) are locally finite, then the radicals \(\alpha(\Gamma,\mathfrak G)\) and \(\beta(\Gamma,\mathfrak G)\) coincide. In the case of a faithful representation, moreover, the radical of the representation is an abstract locally nilpotent group.

Here the inclusion \(\alpha(\Gamma,\mathfrak G)\subset \beta(\Gamma,\mathfrak G)\) follows from the following remark: if the representation \(f\) is algebraic, the group \(\mathfrak G\) is locally finite, and \(\Gamma\) is weakly stable with respect to \(\mathfrak G\), then \(\Gamma\) is locally stable.

The second assertion of the theorem follows from the first and from \((^{3})\). All the preceding arguments naturally carry over to the case where the group \(\mathfrak G\) is considered together with some domain \(\Omega\) of distributive operators.

It is clear that in this case the subgroup is an \(\Omega\)-admissible subgroup, and the automorphism is an operator automorphism. Let us note, in particular, that the following analogue of Theorem 3 holds:

Theorem 4. Let a representation of a locally finite group \(\Gamma\) with respect to a linear space \(\mathfrak G\) be given. Then
\[ \alpha(\Gamma,\mathfrak G)=\beta(\Gamma,\mathfrak G). \]
If the representation is faithful, then \(\alpha(\Gamma,\mathfrak G)\) is a locally nilpotent group.

It is easy to observe that the radical \(\alpha(\Gamma,\mathfrak G)\) is, in a known sense, an analogue of the Jacobson radical from ring theory \((^4)\). Let us define one analogue of the Levitzki radical.

As is known, if \(K\) is the ring of endomorphisms of an abelian group \(\mathfrak G\), the external and internal local nilpotence of such a ring are equivalent notions. Recall that \(K\) is called externally nilpotent if in the group \(\mathfrak G\) there is a series of \(K\)-admissible subgroups \(\mathfrak G_i\), \(\mathfrak G_i \supset \mathfrak G_{i+1}\), \(\mathfrak G_0=\mathfrak G\), \(\mathfrak G_n=0\), such that \(\mathfrak G_i a \subset \mathfrak G_{i+1}\) for every \(i\) and for every \(a\in K\). \(K\) is externally locally nilpotent if every subring of \(K\) with a finite number of generators is externally nilpotent.

Let now a representation of a group \(\Gamma\) with respect to a group \(\mathfrak G\) (not necessarily commutative) be given. By analogy with rings, the group \(\Gamma\) will be called externally locally nilpotent if every subgroup of \(\Gamma\) having a finite number of generators is nilpotent with respect to \(\mathfrak G\). In the case of a faithful representation, external local nilpotence of \(\Gamma\) implies its internal local nilpotence \((^{5,6})\). The converse assertion, unlike for rings, is false. Denote by \(\gamma(\Gamma,\mathfrak G)\) the normal divisor of the group \(\Gamma\) generated by all externally locally nilpotent normal divisors of \(\Gamma\). The analogy between \(\gamma(\Gamma,\mathfrak G)\) and the Levitzki radical of the ring of endomorphisms is evident.

Theorem 5. The radical \(\gamma(\Gamma,\mathfrak G)\) is an externally locally nilpotent group.

The proof of this theorem can be obtained by the methods of work \((^1)\), taking into account that \(\gamma(\Gamma,\mathfrak G)\) is contained in the external radical of the group \(\Gamma\), defined in \((^7)\).

From the definitions the inclusion follows immediately
\[ \gamma(\Gamma,\mathfrak G)\subset \beta(\Gamma,\mathfrak G). \]

Under certain restrictions these two radicals coincide.

Theorem 6. If \(\Gamma\) is an \(LM\)-group, then the radical \(\gamma(\Gamma,\mathfrak G)\) belongs to the radical \(\alpha(\Gamma,\mathfrak G)\).

We shall next consider the case when there is a direct connection between the radical \(\gamma(\Gamma,\mathfrak G)\) of the group of automorphisms and the Levitzki radical of the endomorphism ring.

Let \(\mathfrak G\) be an abelian group, \(K\) some ring of endomorphisms of the group \(\mathfrak G\), containing the identity \(e\), which is the identity operator, and let \(\Gamma\) be the totality of all invertible elements of \(K\). \(\Gamma\) is a group of automorphisms of the group \(\mathfrak G\). Denote by \(L(K)\) the Levitzki radical of the ring \(K\).

The following inclusion is easily verified:
\[ e+L(K)\subset \gamma(\Gamma,\mathfrak G). \]
Under certain additional restrictions this inclusion can be replaced by equality. This will be the case, for example, if \(\Gamma\) generates the whole ring \(K\) and, in addition, one of the following conditions is satisfied: a) the group \(\gamma(\Gamma,\mathfrak G)\) is externally nilpotent; b) \(\Gamma\) is an \(LM\)-group.

The question of the exact conditions for the coincidence of \(e+L(K)\) and \(\gamma(\Gamma,\mathfrak G)\) remains unresolved for the time being.

Let us also note that in the situation under consideration there is likewise the inclusion
\[ e+D(K)\subset \alpha(\Gamma,\mathfrak G). \]
Here by \(D(K)\) we have denoted the Jacobson radical of the ring \(K\).

We now turn to some applications for the special case where the group \(\Gamma\) is taken to be the group \(\mathfrak G\) of inner automorphisms of the group \(\mathfrak G\). First of all, let us agree to denote by \(\alpha(\mathfrak G)\), \(\beta(\mathfrak G)\), and \(\gamma(\mathfrak G)\), respectively, the full inverse images in \(\mathfrak G\) of the radicals \(\alpha(\mathfrak G,\mathfrak G)\), \(\beta(\mathfrak G,\mathfrak G)\), and \(\gamma(\mathfrak G,\mathfrak G)\). It has already been noted that \(\beta(\mathfrak G)\) coincides with the locally nilpotent radical of the group \(\mathfrak G\).

From Theorem 3 and the remark concerning the radical \(\beta(\mathfrak G)\), the following follows immediately.

Theorem 7. In a locally finite group the locally nilpotent radical coincides with the radical \(\alpha(\mathfrak G)\).

This theorem shows, in particular, that in locally finite groups the factor group by the locally nilpotent radical has a good property: it is a subdirect product of automorphism groups induced by the group on the factors of its chief systems. Theorem 7 is known for finite groups. In a recent paper of P. Hall \((^8)\) an analogous theorem is proved for some other classes of groups.

From Theorem 2 for \(LM\)-groups we obtain the inclusion \(\beta(\mathfrak G)\subset \alpha(\mathfrak G)\). The same inclusion also holds in the case where the group \(\mathfrak G\) is an extension of a \(ZA\)-group by means of an \(LM\)-group. Indeed, it is easy to show that in this case, for any subgroup \(H\subset \mathfrak G\) having a finite number of generators, the intersection \(H\cap \beta(\mathfrak G)\) is a \(ZA\)-group. Our assertion now follows immediately from the local theorem.

Concerning the radical \(\gamma(\mathfrak G)\), let us note that it contains the Fitting radical, i.e., the subgroup generated by all nilpotent normal divisors of the group. Probably \(\gamma(\mathfrak G)\) is wider than the Fitting radical.

Received
9 XII 1961

References

\(^1\) B. I. Plotkin, Sibirsk. Mat. Zh., 2, No. 1, 100 (1961).
\(^2\) A. I. Mal’tsev, Uch. Zap. Ivanovsk. Ped. Inst., 10, 10 (1956).
\(^3\) V. G. Vilyatser, DAN, 131, No. 4, 728 (1960).
\(^4\) N. Jacobson, Structure of Rings, IL, 1961.
\(^5\) P. Hall, Illinois J. Math., 2, No. 4 B, 787 (1958).
\(^6\) B. I. Plotkin, V. G. Vilyatser, DAN, 134, No. 3, 529 (1960).
\(^7\) B. I. Plotkin, DAN, 140, No. 5, 1019 (1961).
\(^8\) P. Hall, Proc. London Math. Soc., 11, No. 42, 327 (1961).