S. A. Chunikhin

On indexials of finite groups

(Presented by Academician I. M. Vinogradov, 21 VII 1960)

§ 1. In papers \((^{1-4})\) we proposed and developed the method of “indexials” for finding subgroups in finite groups, and with its aid a general theorem on the existence of subgroups was obtained, embracing as special cases many principal earlier results in this area.

In the present paper, in order to strengthen this theorem, the notions of a regular indexial, an extension of an indexial, and a measure of solvability of a finite group are introduced; with their aid, assertions 5) and 6) of the theorem of the article \((^4)\) are refined, and Theorem 2 of \((^{1,2})\) is generalized.

The main result of the present paper is Theorem 5.

§ 2. We shall use the definitions and notation introduced in \((^3)\), as well as the following:

1) The sequence \(\omega\) and the factor groups \(\mathfrak F_i/\mathfrak G_i,\ i \in \omega\), from article \((^3)\) will be called respectively the basis and the factors of the indexial \((h)_{R,f}\).

2) Let \(I\) be the sequence of indices of some composition (chief) series of a finite group \(\mathfrak G\), and let \(I_{\mathrm{pr}}\) be the subsequence of all those indices from \(I\) which are prime numbers; then the number \(\bar I_{\mathrm{pr}}\) (for the meaning of the symbol \(\bar I_{\mathrm{pr}}\), see \((^3)\)) will be called the measure of solvability (strong solvability) of the group \(\mathfrak G\).

3) An indexial \((h)_{R,f}\) will be called regular if it satisfies the following two requirements: a) the group \(\mathfrak G\) has at least one subgroup \(\mathfrak H\) of order \(h\), contained in \(\mathfrak G_{\beta-1}\); b) for each \(i \in \omega\) one has \(\mathfrak F_i = [\mathfrak H \cap \mathfrak G_{i-1}]\mathfrak G_i\). In this case the subgroup \(\mathfrak H\) will be called a suitable subgroup of the indexial (for the indexial) \((h)\).

4) Let \(\omega\) be a common basis; \(f_\beta,\ f_{\beta+1}, \ldots, f_\omega\) and \(\varphi_\beta = c_\beta f_\beta,\ \varphi_{\beta+1}=c_{\beta+1}f_{\beta+1}, \ldots,\ \varphi_\omega=c_\omega f_\omega\) \((c_\beta, c_{\beta+1}, \ldots, c_\omega\) are natural numbers, of which \(c_\beta=1)\) are the components respectively of the indexials \((h)_{R,f}\) and \((ch)_{R,\varphi}\), \(c=c_\beta c_{\beta+1}\cdots c_\omega\), of the group \(\mathfrak G\). If for each \(i \in \omega\) the factor \(\mathfrak C_i/\mathfrak G_i\) of order \(c_i f_i\) of the indexial \((ch)_{R,\varphi}\) is an extension in \(\mathfrak G_{i-1}/\mathfrak G_i\) of the corresponding factor \(\mathfrak F_i/\mathfrak G_i\) of the indexial \((h)_{R,f}\) by means of a group \(\mathfrak C_i^*\) of order \(c_i\), then the indexial \((ch)_{R,\varphi}\) will be called an extension of the indexial \((h)_{R,f}\). The number \(c\) will be called the measure of extension of the indexial \((h)\).

5) An extension \((ch)_{R,\varphi}\) of the indexial \((h)_{R,f}\) will be called: a) solvable (special) if each group \(\mathfrak C_i^*,\ i \in \omega\), is solvable (special); b) arithmetically closed if \(\Pi(c_i)\subseteq \Pi(f_\beta f_{\beta+1}\ldots f_{i-1})\) for all \(i=\beta+1,\ \beta+2,\ldots,\omega\); c) specially arithmetically closed (s.a.c.) if \((ch)_{R,\varphi}\) is a special and arithmetically closed extension of the indexial \((h)_{R,f}\); d) regular if \((ch)_{R,\varphi}\) is a regular indexial; e) regular s.a.c. if \((ch)_{R,\varphi}\) is a regular and s.a.c. extension of the indexial \((h)_{R,f}\); f) trivially s.a.c. if \(c_i=1\) for each \(i \in \omega\).

6) An indexial admitting no s.a.z. extensions other than the trivial one shall be called s.a.z. maximal.

7) If \(f_i^*,\, i\in\omega\), is the solubility measure of the factor \(\mathfrak F_i/\mathfrak G_i\) of the indexial \((h)_{R,f}\), then

\[ h^*=f_\beta^* f_{\beta+1}^*\cdots f_\omega^* \]

shall be called the solubility measure of \((h)_{R,f}\).

1. We present the results obtained by us.

Theorem 1. If \((h)_{R,f}\) is a proper indexial and \(\mathfrak H\) is its corresponding subgroup, then the factors of the series

\[ \mathfrak H=\mathfrak H\cap\mathfrak G_{\beta-1}\supseteq \mathfrak H\cap\mathfrak G_\beta\supseteq\cdots\supseteq \mathfrak H\cap\mathfrak G_\omega=\mathfrak E \]

of normal divisors of \(\mathfrak H\) are respectively isomorphic to the factors of the indexial \((h)_{R,f}\).

Theorem 2. If \((h)_{R,f}\) is a proper indexial and \(\mathfrak H\) is its corresponding subgroup, then the solubility measure of \(\mathfrak H\) is equal to the solubility measure of \((h)_{R,f}\).

Theorem 3. If \((ch)_{R,\varphi}\) is a soluble extension of the indexial \((h)_{R,f}\), then the solubility measure of \((ch)_{R,\varphi}\) is equal to \(ch^*\), where \(c\) is the measure of the extension and \(h^*\) is the solubility measure of \((h)_{R,f}\).

Theorem 4. If \((ch)_{R,\varphi}\) is a proper and soluble extension of the indexial \((h)_{R,f}\), then the solubility measure of the corresponding subgroup \(\mathfrak H\) of the indexial \((ch)_{R,\varphi}\) is equal to \(ch^*\), where \(c\) is the measure of the extension and \(h^*\) is the solubility measure of \((h)_{R,f}\).

Theorem 5. Every indexial has at least one proper s.a.z. extension.

Theorem 6. Every s.a.z. maximal indexial is proper.

Institute of Mathematics and Computer Engineering
Academy of Sciences of the BSSR

Received
18 VII 1960

REFERENCES

1. S. A. Chunikhin, DAN, 121, No. 2, 243 (1958).
2. S. A. Chunikhin, Izv. Vyssh. uchebn. zaved., ser. matem., No. 1 (14), 227 (1960).
3. S. A. Chunikhin, DAN, 126, No. 2, 284 (1959).
4. S. A. Chunikhin, DAN, 128, No. 6, 1135 (1959).