S. A. Chunikhin

ON SYSTEMS OF NONSPECIAL SUBGROUPS OF FINITE GROUPS

(Presented by Academician A. I. Mal’cev on 28 V 1961)

§ 1. In the present paper we continue the study of systems of nonspecial subgroups introduced by us earlier in \((^1)\). In doing so, we introduce the general notion of an oriented subgroup and of an oriented \(\Pi\)-system, generalizing the notion, introduced in \((^1)\), of an oriented \(\Pi S\)-system, and prove a theorem on the existence of oriented \(\Pi\)-systems in nonspecial groups (Theorem 1). We also investigate the influence of the dispersiveness of a group on the existence in the group of systems of subgroups of a definite kind (Theorems 2, 3, and 4). Along the way we obtain a theorem on the factorization of solvable groups, based on the decomposition, introduced by us, of the set of prime divisors of the order of a group into the so-called \(S\)-classes (Theorem 6). We also introduce indecomposable \(\Pi\)-systems and establish a theorem on their existence.

§ 2. We shall use the notation and definitions introduced in \((^1)\), as well as the following. \(\Pi(n)\), where \(n\) is a natural number, is the set of all prime divisors of \(n\). \(\mathfrak G\) is a finite group of order \((\mathfrak G)\); when \((\mathfrak G)=1\), we put \(\mathfrak G=\mathfrak E\). \(\Pi\) is some empty or nonempty subset of the set \(\Pi((\mathfrak G))\); \(k\) is the number of elements of \(\Pi\). A \(\Pi\)-group is a finite group \(\mathfrak G\) for which \(\Pi=\Pi((\mathfrak G))\). By a \(\Pi\)-system of subgroups of the group \(\mathfrak G\) (more briefly, a \(\Pi\)-system of the group \(\mathfrak G\)) we shall mean, for nonempty \(\Pi\), such a set \(K\) of pairwise nonisomorphic subgroups of the group \(\mathfrak G\) that there exists a one-to-one mapping \(\varphi\) of the set \(\Pi\) onto the set \(K=\varphi(\Pi)\), under which the image \(\varphi(p_i)\) of each \(p_i \in \Pi\) is a \(p_i d\)-subgroup of \(\mathfrak G\). We shall call the function \(\varphi\) the defining function of the system \(K\). For empty \(\Pi\), by definition, we shall regard the empty set as a \(\Pi\)-system \(\varphi(\Pi)\) of the group \(\mathfrak G\). A \(\Pi\)-system for \(\Pi=\Pi((\mathfrak G))\) will simply be called a system. If \(p_i \in \Pi\), then a \(p_i d\)-subgroup \(\mathfrak H\) of the group \(\mathfrak G\) will be called \(p_i\)-oriented relative to the group \(\mathfrak G\) if \(\mathfrak H\) inherits from \(\mathfrak G\) the properties of being non-\(p_i\)-nilpotent and non-\(p_i\)-decomposable. A \(\Pi\)-system \(\varphi(\Pi)\) of the group \(\mathfrak G\) will be called an oriented \(\Pi\)-system of the group \(\mathfrak G\) if \(\Pi\) is empty or if for every \(p_i \in \Pi\) the subgroup \(\varphi(p_i)\) is \(p_i\)-oriented relative to \(\mathfrak G\). If the group \(\mathfrak G\) is a group of type \(S\) or a direct product of two groups of coprime orders, one of which is of type \(S\) and the other cyclic of prime order, then \(\mathfrak G\) will be called a group of type \(SC\) (or an \(SC\)-group). A \(\Pi\)-system (oriented \(\Pi\)-system) will be called: a \(\Pi S\)-system (oriented \(\Pi S\)-system) if \(\Pi\) is empty or if for every \(p_i \in \Pi\) the subgroup \(\varphi(p_i)\) is of type \(S\); a \(\Pi SC\)-system (oriented \(\Pi SC\)-system) if \(\Pi\) is empty or if for every \(p_i \in \Pi\) the subgroup \(\varphi(p_i)\) is of type \(SC\). When \(\Pi=\Pi((\mathfrak G))\) we shall omit the letter \(\Pi\) in these designations. A dispersive group is a group \(\mathfrak G\) that is \(\sigma\)-dispersive \((^2)\), where \(\sigma\) is the set \(\Pi((\mathfrak G))\) endowed with some ordering (cf. Ore \((^3)\)). By a group of type \(Z_p^{(1)}\) we shall mean a \(p\)-indecomposable group whose order is divisible by no more than three distinct primes and which contains no more than one class of nontrivial isomorphic \(p\)-indecomposable …

subgroups. A $\Pi$-set $\varphi(\Pi)$ will be called an indecomposable $\Pi$-set if $\Pi$ is empty or if, for every $p_i \in \Pi$, the subgroup $\varphi(p_i)$ is of type $Z^{(1)}_{p_i}$.

It is obvious that the class of groups of type $SC$ is the set of all groups of type $S_p$, as $p$ ranges over all prime numbers.

§ 3. We present the results obtained by us concerning sets of subgroups.

Theorem 1. A nonspecial group $\mathfrak{G}$ has at least one oriented $\Pi SC$-set containing no fewer than $k-1$ subgroups.

Theorem 2. If $\mathfrak{G}$ has no $\Pi SC$-set of nontrivial subgroups containing $k$ subgroups, then $\mathfrak{G}$ is a dispersion group.

Theorem 3. Let $\mathfrak{G}\ne \mathfrak{E}$ not decompose into a direct product of nontrivial subgroups of pairwise relatively prime orders. Then: 1) if $\mathfrak{G}$ is not a dispersion $\Pi$-group, then it has at least one $\Pi S$-set containing $k$ subgroups; 2) if $\mathfrak{G}$ is a dispersion $\Pi$-group, then it has at least one $\Pi S$-set containing $k-1$ subgroups.

Theorem 4. Suppose that for each $p_i \in \Pi$ the group $\mathfrak{G}$ is indecomposable, and suppose that in the decomposition of $\mathfrak{G}$ into a direct product of nontrivial subgroups of pairwise relatively prime orders, each of which is already indecomposable in this same way, there are altogether $l$ direct factors that are dispersion $\Pi$-groups.

Then $\mathfrak{G}$ has at least one $\Pi S$-set containing $k-l$ subgroups.

Theorem 5. If for every $p_i \in \Pi$ the group $\mathfrak{G}$ is not $p_i$-decomposable and if $\mathfrak{G}$ is not of type $S$, then it has at least one indecomposable $\Pi$-set containing $k$ subgroups.

§ 4. For the proof of Theorem 3 (and of the following Theorem 4 derived from it), we use the notion, introduced below, of $S$-connectedness of the prime divisors of the order of a group and the resulting factorization of soluble groups (Theorem 6). We give the main points of this method.

Let a symbol of the form $\mathfrak{S}_i(p,q)=\mathfrak{S}_i(q,p)$ denote some subgroup of type $S$, whose order is generated by the prime numbers $p$ and $q$. Then the prime divisors $p$ and $q$ of the order of the group $\mathfrak{G}$ will be called $S$-connected in the group $\mathfrak{G}$ by means of the sequence of prime numbers

\[ p^{(1)},\ p^{(2)},\ldots,\ p^{(t+1)}, \tag{1} \]

if in $\mathfrak{G}$ there exists a sequence of subgroups of type $S$ that can be written in the form

\[ \mathfrak{S}_1(p^{(1)},p^{(2)}),\ \mathfrak{S}_2(p^{(2)},p^{(3)}),\ \mathfrak{S}_3(p^{(3)},p^{(4)}),\ldots,\mathfrak{S}_t(p^{(t)},p^{(t+1)}), \tag{2} \]

where $p^{(1)}=p$ and $p^{(t+1)}=q$, $t\ge 1$, and both sequence (1) and sequence (2) may have repetitions. We shall call sequence (2) an $S$-chain passing through sequence (1).

Lemma. If $\Pi$ is the set formed by all members of sequence (1) except the last, then $\mathfrak{G}$ has a $\Pi S$-set containing $k$ subgroups.

Let now $M$ be the set of all those elements of $\Pi((\mathfrak{G}))$ with respect to each of which $\mathfrak{G}$ is not decomposable${}^{(1)}$. By means of Theorem 2 from ${}^{(1)}$ it is not difficult to see that, if $M$ is nonempty, then the property of $S$-connectedness partitions $M$ into classes $M_1,M_2,\ldots,M_r$ of mutually $S$-connected numbers, which we shall call the $S$-classes of the group $\mathfrak{G}$.

By $\Pi_i$ we shall denote some subset of the set $\Pi((\mathfrak{G}))$, and by $\mathfrak{G}_{\Pi_i}$ the set of all $\Pi_i$-elements of $\mathfrak{G}$.

Theorem 6. If $\mathfrak{G}\ne \mathfrak{E}$ is a soluble group, then

\[ \mathfrak{G}=\mathfrak{G}_{\Pi_1}\times \mathfrak{G}_{\Pi_2}\times\cdots\times \mathfrak{G}_{\Pi_\omega}, \]

where \(\Pi_i,\ i=1,2,\ldots,\omega,\) either consists of only one prime number, or is an \(S\)-class of the group \(\mathfrak G\), and \(\mathfrak G_{\Pi_i}\) is a subgroup which is no longer decomposable into a direct product of nontrivial subgroups of pairwise coprime orders.

§ 5. Theorem 4 raises the lower bound for the number of subgroups in \(\Pi S\)-sets of a group, found by V. I. Sergienko \((^4)\).

Gomel Branch
of the Institute of Mathematics and Computer Technology
of the Academy of Sciences of the BSSR

Received
17 V 1962

CITED LITERATURE

\(^1\) S. A. Chunikhin, DAN, 118, No. 4, 654 (1958).
\(^2\) R. Baer, Arch. Math., 9, 7 (1958).
\(^3\) O. Ore, Duke Math. J., 5, 431 (1939).
\(^4\) V. I. Sergienko, DAN, 146, No. 6 (1962).