Abstract
Full Text
MATHEMATICS
K. K. SHCHUKIN
ON THE THEORY OF RADICALS IN GROUPS
(Presented by Academician P. S. Aleksandrov, 11 X 1961)
The present note is a direct continuation of the work of A. G. Kurosh (¹) on the theory of radicals in groups.
§ 1. Let \(\Sigma\) denote, everywhere where this is not specifically stated otherwise, an arbitrary abstract class of groups containing the unit group.
In an arbitrary group \(G\) the following conditions are equivalent:
(b) The product of invariant \(\Sigma\)-subgroups of \(G\) is a \(\Sigma\)-group.
(b\(_1\)) The subgroup of the group \(G\) generated by the \(\Sigma\)-subgroups attainable in \(G\) is a \(\Sigma\)-group.
Obviously, (b) follows from (b\(_1\)). The converse assertion is easily derived from the following fact.
Lemma 1. If \(\Sigma\) is a class of groups with condition (b), then every attainable \(\Sigma\)-subgroup of a group \(G\) generates an invariant \(\Sigma\)-subgroup in this group.
From this lemma and axiom II.21 of (¹) we obtain also the following
Corollary 1. If \(R\) is a radical class, then a group \(G\) in which every homomorphic image different from the unit group contains a nonunit attainable \(R\)-subgroup is an \(R\)-group.
It is also easy to see that axiom I.2 from (¹) for \(\Sigma\) is equivalent to condition (b) and, consequently, to (b\(_1\)). In the case of a radical class \(R\) this leads to the following:
In every group \(G\) its \(R\)-radical coincides with the subgroup generated by all attainable \(R\)-subgroups in \(G\).
Let us note, finally, that the system of radicality axioms I.1— I.3 of A. G. Kurosh (¹) for a class of groups \(R\) is equivalent to the following:
I.a. A homomorphic image of an \(R\)-group is an \(R\)-group.
I.b. The product of invariant \(R\)-subgroups of a group \(G\) is an \(R\)-group.
I.c. An extension of an \(R\)-group by means of an \(R\)-group is an \(R\)-group.
§ 2. Denote, following (¹), by \(M\) a class of groups satisfying axiom I.a, and by \(R_0(M)\) the minimal radical class containing all \(M\)-groups. Below are given some characteristics of \(R_0(M)\)-groups.
Theorem 1. A group \(G\) is an \(R_0(M)\)-group if and only if every homomorphic image of it different from the unit group contains a nonunit attainable \(M\)-subgroup.
The necessity of this assertion is trivial in view of (¹), and the sufficiency is easily derived from Corollary 1 of the preceding paragraph.
A. G. Kurosh established (¹) that the groups of \(\alpha\)-th degree over \(M\), where \(\alpha\) is an arbitrary ordinal number, and only they, constitute the class \(R_0(M)\).
Theorem 2. Let \(\alpha\) be an infinite ordinal number. Then every group of \(\alpha\)-th degree over \(M\) has \(\omega_0\)-th degree over \(M\), where \(\omega_0\) is the first infinite ordinal number.
The last result is a consequence of Theorem 1 and the following lemma:
Lemma 2. Let the class of groups \(M\) satisfy condition I.a. Then every attainable \(M\)-subgroup \(A \ne E\) of defect \(d \ne 0\) in a group \(G\) generates in this group an invariant subgroup of \(d\)-th degree over \(M\).
Let us note that the defect of an attainable subgroup is understood here in the sense of R. Baer (²) and is always a nonnegative integer.
As a consequence of Theorem 2 we obtain still another characterization of the class of groups that interests us.
A group \(G\) is an \(R_0(M)\)-group if and only if it has an ascending invariant series with factors having finite degree over \(M\).
§ 3. In the work of R. Baer \((^2)\), nilgroups were studied, i.e., groups generated by their ascendant cyclic subgroups. It is useful to generalize this concept to the case of an arbitrary abstract class \(\Sigma\) containing the identity group.
Definition. We shall call a group \(G\) a \(\Sigma\)-nilgroup if it is generated by its ascendant \(\Sigma\)-subgroups.
From the transitivity of the property of being an ascendant subgroup it is easy to obtain that \(\Sigma\)-nilgroups satisfy condition (b) and, consequently, \((b_1)\). Hence in every group \(G\) there exists a characteristic \(\Sigma\)-nilsubgroup \(\Sigma(G)\), generated by all ascendant \(\Sigma\)-nilsubgroups in \(G\). Moreover, \(\Sigma(G)\) coincides with the subgroup of the group \(G\) generated by all ascendant \(\Sigma\)-subgroups in \(G\). As follows from § 1, in the case of a radical class \(\Sigma\) the subgroup \(\Sigma(G)\) coincides with the \(\Sigma\)-radical of the group \(G\).
If now the class \(M\) satisfies condition I.a, then this condition will obviously also be satisfied by the class of \(M\)-nilgroups. Hence, from Theorem 1, it is clear that \(M\)-nilgroups are contained in the class of \(R_0(M)\)-groups and, generally speaking, properly so \((^2)\).
Construct in an arbitrary group \(G\) an ascending characteristic series
\[ E=M_0(G)\subset M_1(G)\subset \cdots \subset M_\alpha(G)\subset M_{\alpha+1}(G)\subset\cdots \]
according to the rule: \(M(G/M_\alpha(G))=M_{\alpha+1}(G)/M_\alpha(G)\), and at limit positions take, as usual, the union of the preceding terms. There exists an ordinal \(\lambda\) such that \(M_\lambda(G)=M_{\lambda+1}(G)\). It turns out that the subgroup \(M_\lambda(G)\) coincides with the \(R_0(M)\)-radical of the group \(G\). Hence there arises still another characterization of the class \(R_0(M)\):
A group \(G\) is an \(R_0(M)\)-group if and only if it has an ascending characteristic series with factors that are \(M\)-nilgroups.
§ 4. According to Theorem 2, the first infinite ordinal serves as an upper bound for the degree over \(M\) of every \(R_0(M)\)-group. Here we consider the simplest condition under which a natural number serves as such a bound.
Let us first consider the following condition for an abstract class of groups \(\Sigma\).
\((\beta)\) If a group \(A\ne E\), generated by its invariant \(\Sigma\)-subgroups, is a normal divisor of a group \(B\), then \(A\) contains a nonidentity \(\Sigma\)-subgroup invariant in the whole group \(B\).
Lemma 3. If condition \((\beta)\) is fulfilled for \(\Sigma\), then every nonidentity ascendant \(\Sigma\)-subgroup of minimal defect in a group \(G\) is a normal divisor of this group.
From Theorem 1 and Lemma 3 we easily obtain the following assertion.
If the class of groups of degree \(n\) over \(M\), where \(n\ge 1\) is a fixed natural number, satisfies condition \((\beta)\), then the minimal radical class \(R_0(M)\) consists exactly of the groups of degree \(n\) over \(M\).
Let us recall \((^1)\), by the way, that the groups of degree 2 over the class of abelian groups coincide with the \(RJ^*\)-groups \((^3)\), while the subresolvable groups of R. Baer \((^2)\) are exactly the minimal radical class over the class of abelian groups (see Theorem 1). Up to now the question \((^2)\) of the coincidence of these two classes of groups has remained open. It is even more unknown whether abelian groups will satisfy condition \((\beta)\).
Intermediate between (b) and \((\beta)\) for \(\Sigma\) is the following condition (cf. \((^2)\)):
\((\beta_0)\) In every group \(G\) the product of all nontrivial invariant \(\Sigma\)-subgroups contains a nontrivial characteristic \(\Sigma\)-subgroup.
Since in the presence of \((\beta_0)\) condition \((\beta)\) is fulfilled automatically, from the preceding result we obtain:
If the class of groups of the \(n\)-th degree over \(M\), where \(n \geqslant 1\) is a fixed natural number, satisfies condition \((\beta_0)\), then the class \(R_0(M)\) consists of the groups that possess, in every nontrivial homomorphic image, a nontrivial characteristic subgroup of the \(n\)-th degree over \(M\).
R. Baer \((^2)\) posed the question of the fulfillment of \((\beta_0)\) for abelian groups. D. McLain \((^4)\) published an example of an elementary locally finite \(p\)-group which is an \(RJ^*\)-group, thereby giving a negative answer to this question.
§ 5. Recall that a subinvariant subgroup of a group \(G\) is any subgroup of it that is a member of some ascending (in general, transfinite) normal series of the group \(G\).
Let the abstract class of groups \(\Sigma\) be subject to the requirement:
\((*)\) The set-theoretic sum of an ascending normal chain of \(\Sigma\)-subgroups of a group \(G\) is a \(\Sigma\)-group.
Then, in an arbitrary group \(G\), condition (b) for such a \(\Sigma\) is equivalent to the following:
\((b_2)\) The subgroup of the group \(G\) generated by the subinvariant in \(G\) \(\Sigma\)-subgroups is a \(\Sigma\)-group.
It is obvious that (b) follows from \((b_2)\). For the proof of the converse assertion, one uses
Lemma 4. Let the abstract class \(\Sigma\), with condition \((*)\), be subject to requirement (b). Then every subinvariant \(\Sigma\)-subgroup of the group \(G\) generates in this group an invariant \(\Sigma\)-subgroup.
The last lemma and Axiom II.21 from \((^1)\) also allow one to obtain
Corollary 2. If \(R\) is a radical class of groups with condition \((*)\), then a group \(G\) such that in every nontrivial homomorphic image of it there is a nontrivial subinvariant \(R\)-subgroup is an \(R\)-group.
As already noted in § 1, Axiom I.2 from \((^1)\) is equivalent to condition (b) and, consequently, in the presence of \((*)\), to condition \((b_2)\). Hence we obtain:
If a radical class \(R\) satisfies condition \((*)\), then the \(R\)-radical of a group \(G\) coincides with the subgroup generated by all subinvariant in \(G\) \(R\)-subgroups.
Let, as in \((^1)\), \(R(M)\) denote the class of groups possessing an ascending normal series with \(M\)-factors. The class of \(R(M)\)-groups is radical \((^1)\) and, obviously, is subject to requirement \((*)\). Hence, also from Corollary 2, it is easily derived that
Theorem 3. A group \(G\) is an \(R(M)\)-group if and only if every nontrivial homomorphic image of it contains a nontrivial subinvariant \(M\)-subgroup.
In conclusion I express my sincere gratitude to Prof. A. G. Kurosh, under whose supervision this work was carried out.
Received
10 X 1961
REFERENCES
- A. G. Kurosh, DAN, 141, No. 4 (1961).
- R. Baer, Math. Zs., 62, 4, 402 (1955).
- A. G. Kurosh, Group Theory, Moscow, 1953.
- D. H. McLain, Proc. Cambr. Phil. Soc., 50, 4, 641 (1954).