MATHEMATICS

Yu. M. GORCHAKOV

PRIMARILY FACTORIZABLE GROUPS

(Presented by Academician A. I. Mal'cev on 23 IV 1960)

A subgroup \(\mathfrak A\) of a group \(\mathfrak G\) is called complemented in \(\mathfrak G\) if in \(\mathfrak G\) there exists a subgroup \(\mathfrak B\) such that \(\mathfrak A\mathfrak B=\mathfrak G\) and \(\mathfrak A\cap\mathfrak B=1\). The condition that certain subgroups of a group be complemented sometimes determines its structure to a significant extent. For example, N. V. Baeva-Chernikova \((^1,^2)\), studying arbitrary groups in which all subgroups are complemented (following her, we shall call such groups fully factorizable), showed that they are exhausted by certain periodic subgroups of complete direct products of finite groups whose orders are not divisible by squares of prime numbers, and even by complete direct products of specially chosen groups of this kind (called quite primitive by the author \((^1)\)). Fully factorizable groups have been studied completely in \((^2)\).

Groups with various kinds of systems of complemented subgroups under various additional conditions were considered by S. N. Chernikov in \((^3)\); in that work the general problem of studying groups with prescribed systems of complemented subgroups was first posed. In the present paper we solve the problem, proposed to the author by S. N. Chernikov, on the structure of groups with complemented \(p\)-subgroups for every prime \(p\).

1. We shall call a group \(\mathfrak G\) primarily factorizable if all its \(p\)-subgroups are complemented in it for every prime number \(p\).

Let \(\mathfrak P\) be a group of prime order \(p\). A fully factorizable subgroup \(\mathfrak A\) of the holomorph of the group \(\mathfrak P\) will be called fully \(p\)-primitive if it contains the group \(\mathfrak P\). If there is no need to indicate the order of the group \(\mathfrak P\), then the group \(\mathfrak A\) will simply be called fully primitive.

Let
\[ \widetilde{\mathfrak G}=\widetilde{\prod_{\alpha\in\mathfrak M}}\mathfrak G_\alpha \]
be the complete direct product of certain fully primitive groups \(\mathfrak G_\alpha\), and
\[ \mathfrak G=\prod_{\alpha\in\mathfrak M}\mathfrak G_\alpha \]
the direct product of the same groups. Denote by
\[ \widetilde{\mathfrak G}_p=\widetilde{\prod_{\alpha\in\mathfrak M_p}}\mathfrak G_\alpha \]
the complete direct product of all fully \(p\)-primitive (for fixed \(p\)) groups \(\mathfrak G_\alpha\), \(\alpha\in\mathfrak M\).

A set \(\mathfrak N_p\) of components of the elements of a set \(\mathfrak N\) of elements of the group \(\widetilde{\mathfrak G}\) (not necessarily distinct) in the group \(\widetilde{\mathfrak G}_p\) will be called the \(p\)-projection of the set \(\mathfrak N\).

Some set \(\mathfrak A\) of elements of the group \(\widetilde{\mathfrak G}\) will be called a commutative set if all its elements commute with one another. If to each element \(A\in\mathfrak A\) there is put in correspondence some element \(G\) of the commutant \(\mathfrak G'\) of the group \(\mathfrak G\), then the set \(\widetilde{\mathfrak A}\) of elements \(GA\) will be called an almost commutative set of the group \(\widetilde{\mathfrak G}\).

Let \(\mathfrak N\) be some set of elements of the group \(\widetilde{\mathfrak G}\). If all its \(p\)-projections \(\mathfrak N_p\) (for all possible \(p\)) are almost commutative in the groups \(\widetilde{\mathfrak G}_p\), then \(\mathfrak N\) will be called a primary set of elements of the group \(\widetilde{\mathfrak G}\).

Consider the set of functions \(\varphi(n), \psi(n), \ldots\) of a natural argument \(n\), taking natural values.

Suppose that the equality \(\varphi(n)=\psi(n)\) holds for only a finite number of values of the argument \(n\). We shall call such functions finitely linked. A set \(\mathfrak M\) of pairwise finitely linked functions will be called maximal if it is not contained in another set of pairwise finitely linked functions.

1. Consider the complete direct product
   \[ \widetilde{\mathfrak G}=\widetilde{\prod_{(k,l)}}\,\mathfrak G_{kl} \]
   of nonabelian groups \(\mathfrak G_{kl}\) of order \(2p_{kl}\), where \(p_{kl}\) is a prime number, \(k,l=1,2,\ldots\).

In each of the groups \(\mathfrak G_{kl}\) choose two elements \(A_{kl}\) and \(B_{kl}\) of the second order. In the group \(\widetilde{\mathfrak G}\) choose the elements \(A_n\), \(n=1,2,\ldots\), by the following rule: the component of the element \(A_n\) in the group \(\mathfrak G_{ns}\), \(s=1,2,\ldots\), is equal to \(A_{ns}\), and all the remaining components of the element \(A_n\) are equal to the identity, i.e.
\[ A=A_{n1}A_{n2}\ldots A_{ns}\ldots \]

To every function \(\varphi\) from some (fixed) maximal set \(\mathfrak M\) of pairwise finitely linked functions we put in correspondence an element \(B_\varphi\) of the group \(\widetilde{\mathfrak G}\) according to the following rule: the component of the element \(B_\varphi\) of the group \(\widetilde{\mathfrak G}\) in the group \(\mathfrak G_{s,\varphi(s)}\), \(s=1,2,\ldots\), is equal to the element \(B_{s,\varphi(s)}\), and all the remaining components of the element \(B_\varphi\) are equal to the identity, i.e.
\[ B_\varphi=B_{1,\varphi(1)}B_{2,\varphi(2)}\ldots B_{s,\varphi(s)}\ldots \]

By the elements \(A_n\), \(n=1,2,\ldots\), and \(B_\varphi\), \(\varphi\in\mathfrak M\), generate a subgroup \(\mathfrak G\) of the group \(\widetilde{\mathfrak G}\). If \(p_{kl}=p_{rs}\) if and only if \(k=r\) and \(l=s\), then the group \(\mathfrak G\) is a periodic primarily factorizable group, but is not completely factorizable, since the commutant \(\mathfrak G'\) of the group \(\mathfrak G\) is not complemented in \(\mathfrak G\).

Consequently, the class of periodic primarily factorizable groups is broader than the class of completely factorizable groups.

1. The structure of periodic primarily factorizable groups is described by the following two propositions.

Theorem 1. A periodic subgroup \(\mathfrak A\) of the complete direct product
\[ \widetilde{\mathfrak G}=\widetilde{\prod_{\alpha\in\mathfrak M}}\,\mathfrak G_\alpha \]
of completely primitive groups \(\mathfrak G_\alpha\) is primarily factorizable if and only if each of its \(p\)-projections \(\mathfrak A_p\) is completely factorizable.

Theorem 2. A periodic primarily factorizable group \(\mathfrak H\) is isomorphic to some subgroup \(\widetilde{\mathfrak H}\), generated by a primary set of elements of the complete direct product
\[ \widetilde{\mathfrak G}=\widetilde{\prod_{\alpha\in\mathfrak M}}\,\mathfrak G_\alpha \]
of completely primitive groups \(\mathfrak G_\alpha\).

A periodic subgroup \(\widetilde{\mathfrak H}\) of the group \(\widetilde{\mathfrak G}\), generated by an arbitrary primary set of elements of the group \(\widetilde{\mathfrak G}\), is primarily factorizable.

I express my gratitude to S. N. Chernikov for his guidance in carrying out the present work.

Received
20 IV 1960

CITED LITERATURE

¹ N. V. Baeva, DAN, 97, No. 5, 877 (1953). ² N. V. Chernikova, Matem. sborn., 39, 273 (1956). ³ S. N. Chernikov, Matem. sborn., 35, issue 1, 93 (1954).