Reports of the Academy of Sciences of the USSR

1. Vol. 147, No. 5

MATHEMATICS

S. A. CHUNIKHIN

ON THE DECOMPOSITION OF \(\Pi\)-SOLVABLE GROUPS INTO A DIRECT PRODUCT OF SUBGROUPS

(Presented by Academician I. M. Vinogradov on 23 VI 1962)

§ 1. In papers \((^{4-6})\) we introduced and, to a certain extent, investigated \(p\)-solvable and the more general \(\Pi\)-solvable groups, which subsequently also found application among other authors (P. Hall, G. Higman, B. Huppert, and others). In the present paper, which adjoins the note \((^2)\), a method is established for decomposing \(\Pi\)-solvable groups into a direct product of subgroups constructed with the aid of the concept of \(S\)-connectedness of prime divisors of the order of a group (cf. \((^2)\)). At the same time Theorem 6 of \((^2)\) is generalized.

§ 2. We use the following definitions and notation (cf. \((^1,^2)\)). \(\mathfrak G\) is a certain finite group of order \((\mathfrak G)\); \(\mathfrak E\) is the identity group; if \(n\) is a natural number, then \(\Pi(n)\) is the set of all distinct prime divisors of \(n\); \(\Pi\) is some empty or nonempty subset of the set \(\Pi((\mathfrak G))\); a divisor \(d\) of the number \((\mathfrak G)\) will be called a \(\Pi\)-divisor of \((\mathfrak G)\) if \(\Pi(d)\subseteq\Pi\); a \(\Pi\)-subgroup (\(\Pi\)-element) of the group \(\mathfrak G\) is a subgroup (element) whose order is a \(\Pi\)-divisor of \((\mathfrak G)\); a \(\Pi\)-subgroup whose order is the greatest \(\Pi\)-divisor of \((\mathfrak G)\) will be called a \(\Pi\)-Sylow subgroup of \(\mathfrak G\); \(\mathfrak G_{\Pi}\) is the set of all \(\Pi\)-elements of the group \(\mathfrak G\); \(p\) and \(q\) are prime numbers; a \(p\)-decomposable group is a finite group that is the direct product of its \(p\)-Sylow subgroup and a \(p\)-Sylow complement; \(\mathfrak P\) is some \(p\)-Sylow subgroup of \(\mathfrak G\); a special group is a finite group in which all Sylow subgroups are invariant; a group of type \(S\) is a finite nonspecial group all of whose nontrivial subgroups are special \((^7)\); the order of a group of type \(S\) has the form \(p^\alpha q^\beta\), \(\alpha>0\), \(\beta>0\) \((^8)\); a symbol of the form \(\mathfrak S_i(p,q)=\mathfrak S_i(q,p)\) will denote some subgroup of type \(S\) and of order of the form \(p^\alpha q^\beta\); a \(pd\)-subgroup is a subgroup whose order is divisible by \(p\).

Definition 1. If in \(\mathfrak G\ne\mathfrak E\) there exists a sequence of subgroups of type \(S\), which can be written in the form
\[ \mathfrak S_1(p^{(1)},p^{(2)}),\quad \mathfrak S_2(p^{(2)},p^{(3)}),\quad \mathfrak S_3(p^{(3)},p^{(4)}),\ldots,\quad \mathfrak S_t(p^{(t)},p^{(t+1)}), \]
where \(p^{(1)}=p\in\Pi((\mathfrak G))\) and \(p^{(t+1)}=q\in\Pi((\mathfrak G))\), then \(p\) and \(q\) will be called \(S\)-connected in the group \(\mathfrak G\), and the indicated chain of subgroups will be called a chain \(S\)-connecting \(p\) and \(q\).

Let now \(M\) be the set of all those \(p\in\Pi((\mathfrak G))\) for which \(\mathfrak G\) is not \(p\)-decomposable. With the aid of Theorem 3 of \((^3)\) it is not difficult to see that, if \(M\) is nonempty, then the property of \(S\)-connectedness divides \(M\) into classes \(M_1,\ldots,\ldots,M_r\) of mutually \(S\)-connected numbers, which we shall call the \(S\)-classes of \(\mathfrak G\).

Definition 2. A subset \(\sigma\) of the set \(\Pi((\mathfrak G))\) will be called an \(S\)-portion of the group \(\mathfrak G\) if \(\sigma\) consists of only one number \(p\) and \(\mathfrak G\) is \(p\)-decomposable, or if \(\sigma\) is an \(S\)-class of \(\mathfrak G\).

Theorem. Let \(\mathfrak G\ne\mathfrak E\) be a \(\Pi\)-solvable group, let \(\sigma_1,\sigma_2,\ldots,\sigma_\mu\), \(\mu\ge 0\), be all the \(S\)-portions of the group \(\mathfrak G\) contained in \(\Pi\), and let

\(\tau\) is the union of all the remaining \(S\)-portions of \(\mathfrak G\). Then:

1) \(\mathfrak G_{\sigma_1}, \mathfrak G_{\sigma_2}, \ldots, \mathfrak G_{\sigma_\mu}, \mathfrak G_\tau\) are subgroups and
\[ \mathfrak G=\mathfrak G_{\sigma_1}\times \mathfrak G_{\sigma_2}\times \ldots \times \mathfrak G_{\sigma_\mu}\times \mathfrak G_\tau; \]
2) the subgroup \(\mathfrak G_{\sigma_i}\), \(i=1,2,\ldots,\mu\), is no longer decomposable into a direct product of nontrivial subgroups of pairwise coprime orders.

Proof. We shall first show that assertion 2) of the theorem follows from assertion 1).

Suppose the contrary: \(\mathfrak G_{\sigma_i}=\mathfrak H=\mathfrak H_1\times \mathfrak H_2\), where \((\mathfrak H_1)\) and \((\mathfrak H_2)\) are relatively prime and greater than 1. Let \(p\in \Pi((\mathfrak H_1))\) and \(q\in \Pi((\mathfrak H_2))\). Since \(\mathfrak G_{\sigma_i}\) is an invariant \(\sigma_i\)-Sylow subgroup of \(\mathfrak G\) and \(\sigma_i\) is an \(S\)-portion of \(\mathfrak G\), \(p\) and \(q\) are \(S\)-connected not only in \(\mathfrak G\), but also in \(\mathfrak G_{\sigma_i}=\mathfrak H\). Therefore, in the chain \(S\)-connecting \(p\) and \(q\), there will be a subgroup \(\mathfrak S_i(p^{(i)},p^{(i+1)})\) of type \(S\) from \(\mathfrak H\), for which \(p^{(i)}\in \Pi((\mathfrak H_1))\) and \(p^{(i+1)}\in \Pi((\mathfrak H_2))\). But then, since \(((\mathfrak H_1),(\mathfrak H_2))=1\), the \(p^{(i)}\)-Sylow subgroup \(\mathfrak S_i\) will lie in \(\mathfrak H_1\), while the \(p^{(i+1)}\)-Sylow subgroup \(\mathfrak S_i\) will lie in \(\mathfrak H_2\). Hence, in view of \(\mathfrak H=\mathfrak H_1\times \mathfrak H_2\), it follows that \(\mathfrak S_i\) is a special group. A contradiction has been obtained.

Now suppose that \(\mathfrak G\) is one of the groups of least order for which assertion 1) of the theorem fails. Since for \(\Pi\) empty \(\mathfrak G=\mathfrak G_\tau\), \(\Pi\) is nonempty. Consequently, \((\mathfrak G)>1\). Let then
\[ \mathfrak G\supset \mathfrak G'\supset \ldots \]
be some composition series of \(\mathfrak G\).

Only the following two cases are possible:

1) The index of \(\mathfrak G'\) in \(\mathfrak G\) is not divisible by any prime from \(\Pi\). Then \(\tau\) is nonempty.

Since \((\mathfrak G')<(\mathfrak G)\), for \(\mathfrak G'\) there exists the required direct decomposition, which, as is not difficult to see, in the case under consideration may be given the form
\[ \mathfrak G'=\mathfrak G_{\sigma_1}'\times \mathfrak G_{\sigma_2}'\times \ldots \times \mathfrak G_{\sigma_\mu}'\times \mathfrak G_{\tau'}', \qquad \text{where } \tau'\subseteq \tau . \]

By Theorem 2 of \((^6)\), in \(\mathfrak G\) there exist \(\tau\)-Sylow subgroups. Let \(\mathfrak H\) be one of them. Let now \(p\in \tau\). Since \(\mathfrak G_{\sigma_i}\), \(1\leq i\leq \mu\), is a characteristic subgroup of \(\mathfrak G'\), \(\mathfrak G_{\sigma_i}\) is invariant in \(\mathfrak G\) and \(\mathfrak P\mathfrak G_{\sigma_i}\) is a subgroup. Since \(p\) does not enter the \(S\)-portion \(\sigma_i\) of the group \(\mathfrak G\), \(\mathfrak P\mathfrak G_{\sigma_i}\) has no \(pd\)-subgroups of type \(S\). Then, by Theorem 3 of paper \((^3)\), \(\mathfrak P\mathfrak G_{\sigma_i}\) will be \(p\)-decomposable:
\[ \mathfrak P\mathfrak G_{\sigma_i}=\mathfrak P\times \mathfrak G_{\sigma_i}. \]

Since \(\mathfrak P\) is an arbitrary \(p\)-Sylow subgroup of \(\mathfrak G\), in view of the fact that \(\mathfrak H\) is a \(\tau\)-Sylow subgroup of \(\mathfrak G\), one may regard \(\mathfrak P\) also as an arbitrary \(p\)-Sylow subgroup of \(\mathfrak H\). Hence, taking into account
\[ \mathfrak P\mathfrak G_{\sigma_i}=\mathfrak P\times \mathfrak G_{\sigma_i}, \]
we see that
\[ \mathfrak H\mathfrak G_{\sigma_i}=\mathfrak H\times \mathfrak G_{\sigma_i},\qquad 1\leq i\leq \mu . \]
But it is obvious that
\[ \mathfrak G=(\mathfrak G_{\sigma_1}\times \mathfrak G_{\sigma_2}\times \ldots \times \mathfrak G_{\sigma_\mu}\times \mathfrak G_{\tau'})\mathfrak H . \]
Hence, taking the preceding equalities into account, we see that
\[ \mathfrak G=\mathfrak G_{\sigma_1}\times \mathfrak G_{\sigma_2}\times \ldots \times \mathfrak G_{\sigma_\mu}\times (\mathfrak G_{\tau'}\mathfrak H), \]
whence
\[ \mathfrak G_{\tau'}\mathfrak H=\mathfrak H=\mathfrak G_\tau . \]
A contradiction has been obtained.

2) The index of \(\mathfrak G'\) in \(\mathfrak G\) is equal to \(p\in \Pi\).

Since \(\mathfrak G'\) is also \(\Pi\)-solvable, and \((\mathfrak G')<(\mathfrak G)\), for \(\mathfrak G'\) there exists the decomposition required by the theorem:
\[ \mathfrak G'=\mathfrak G_{\sigma_1'}'\times \mathfrak G_{\sigma_2'}'\times \ldots \times \mathfrak G_{\sigma_{\mu'}'}'\times \mathfrak G_{\tau''}', \qquad \mu'\geq 0,\quad \tau'\subseteq \tau, \]
and each \(\sigma_i'\subseteq \Pi\), \(i=1,2,\ldots,\mu'\), is contained in some \(S\)-portion of \(\mathfrak G\). Then
\[ \mathfrak G=\mathfrak P\mathfrak G'. \]

Further, two cases are possible.

a) \(\mathfrak G\) is \(p\)-decomposable. If \(\mathfrak G'\) is not a \(pd\)-subgroup, then
\[ \mathfrak G=\mathfrak P\times \mathfrak G_{\sigma_1'}'\times \mathfrak G_{\sigma_2'}'\times \ldots \times \mathfrak G_{\sigma_{\mu'}'}'\times \mathfrak G_{\tau'}' \]
will obviously be the desired decomposition of \(\mathfrak G\). Let \(\mathfrak G'\) be a \(pd\)-group. Since \(\mathfrak G'\) is also \(p\)-decomposable, one of the sets \(\sigma_1',\sigma_2',\ldots,\sigma_{\mu'}'\) (for example, \(\sigma_1'\)) will be of the form \(\{p\}\). Then
\[ \mathfrak G=\mathfrak P\times \mathfrak G_{\sigma_2'}'\times \ldots \times \mathfrak G_{\sigma_{\mu'}'}'\times \mathfrak G_{\tau'}' \]
will obviously be a decomposition of the required form. A contradiction has been obtained.

b) \(\mathfrak G\) is not \(p\)-decomposable. Then, by Theorem 3 of \((^3)\), the number \(p\) is \(S\)-connected with some elements of \(\Pi((\mathfrak G'))\). Let \(\rho\) be the union of all those

sets from the collection \(\sigma_1', \sigma_2', \ldots, \sigma_\mu', \tau'\), which contain at least one element \(S\)-connected with \(\rho\). In the present case \(\rho\) is nonempty.

It is clear that \(\mathfrak{G}_\rho'\) will be the product of the elements of some nonempty collection \(\mathfrak{M}\) of direct factors of the decomposition

\[ \mathfrak{G}'=\mathfrak{G}_{\sigma_1'}' \times \mathfrak{G}_{\sigma_2'}' \times \cdots \times \mathfrak{G}_{\sigma_\mu'}' \times \mathfrak{G}_{\tau'}'. \]

If outside \(\mathfrak{M}\) there are no longer any factors of this decomposition, then \(\mathfrak{G}=\mathfrak{P}\mathfrak{G}_\rho'\). It is clear that then \(\Pi((\mathfrak{P}\mathfrak{G}_\rho'))\) will be either the union \(\tau\) of the \(S\)-portions of the group \(\mathfrak{G}\) not entering into \(\Pi\), or the \(S\)-portion \(\sigma=\Pi\) of the group \(\mathfrak{G}\). Then either \(\mathfrak{G}=\mathfrak{G}_\tau\), or \(\mathfrak{G}=\mathfrak{G}_\sigma\). We have obtained a contradiction.

Let outside \(\mathfrak{M}\) there exist factors \(\mathfrak{G}_{\omega_1}', \mathfrak{G}_{\omega_2}', \ldots, \mathfrak{G}_{\omega_\lambda}'\) of the above decomposition of \(\mathfrak{G}'\) (clearly, \(\omega_1, \omega_2, \ldots, \omega_\lambda\) coincide with some of the sets \(\sigma_1', \sigma_2', \ldots, \sigma_\mu', \tau'\)).

Since \(\omega_1, \omega_2, \ldots, \omega_\lambda\) have no elements in common with the set \(\rho\), as in case 1), we are convinced that

\[ \mathfrak{P}\mathfrak{G}_{\omega_i}'=\mathfrak{P}\times \mathfrak{G}_{\omega_i}', \quad i=1,2,\ldots,\lambda. \]

Then

\[ \mathfrak{G}=\mathfrak{P}\mathfrak{G}'=(\mathfrak{P}\mathfrak{G}_\rho')\times \mathfrak{G}_{\omega_1}'\times \mathfrak{G}_{\omega_2}'\times \cdots \times \mathfrak{G}_{\omega_\lambda}'. \]

If \(\tau'\) is nonempty and enters into \(\rho\), then put \(\tau=\Pi((\mathfrak{P}\mathfrak{G}_\rho'))\); then all \(\omega_i\), \(i=1,2,\ldots,\lambda\), will be \(S\)-portions of \(\mathfrak{G}\) entering into \(\Pi\).

If, however, \(\tau'\) is nonempty and coincides with one of the sets \(\omega_1, \omega_2, \ldots, \omega_\lambda\) (or is empty), then the remaining ones among them (or all of them), as well as \(\Pi((\mathfrak{P}\mathfrak{G}_\rho'))\), will clearly be \(S\)-portions of \(\mathfrak{G}\) entering into \(\Pi\). In all cases the decomposition of \(\mathfrak{G}\) obtained will be of the required type. A contradiction has been obtained. The theorem is proved.

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

Received
20 VI 1962

REFERENCES

\({}^{1}\) S. A. Chunikhin, DAN, 118, No. 4, 654 (1958). \({}^{2}\) S. A. Chunikhin, DAN, 146, No. 6 (1962). \({}^{3}\) I. K. Chunikhin, S. A. Chunikhin, Matem. sborn., 15 (57), 2, 325 (1944). \({}^{4}\) S. A. Chunikhin, DAN, 55, No. 6, 481 (1947). \({}^{5}\) S. A. Chunikhin, Matem. sborn., 25 (67), 3, 321 (1949). \({}^{6}\) S. A. Chunikhin, DAN, 73, No. 1, 29 (1950). \({}^{7}\) S. A. Chunikhin, Matem. sborn., 40, 1, 39 (1933). \({}^{8}\) O. Yu. Schmidt, Matem. sborn., 31, 366 (1924).