V. P. Gromyko

On a New Criterion for the Speciality of \(\pi d\)-Groups with a Given Number of Classes of Unattainable Isoordinal \(\pi d\)-Subgroups

(Presented by Academician A. I. Mal’tsev on 24 I 1963)

§ 1. In the present paper we mainly use the notation, concepts, and definitions of our preceding notes \((^{1-3})\). In addition, we shall use the terminology of papers \((^{4,5})\), concerning isoordinal subgroups and classes of isoordinal attainable and unattainable subgroups. The number of classes of unattainable isoordinal \(\pi d\)-subgroups \((^{6})\) we shall agree to denote by \(r\), and the number of distinct prime \(\pi\)-divisors of the order of the group \(G\) by \(t\).

In paper \((^{7})\) a theorem had already been obtained on the solvability of a group \(G\) for which \(r=t-1\), and in paper \((^{5})\)—on the \(\pi\)-solvability of \(G\) satisfying the equality \(r=t\), and on the \(\pi\)-separability of \(G\) with the relation \(r=t+1\). The latter results generalized and strengthened Theorems 2 and 3 \((^{1}), (^{8-10})\) and Theorems 4 and 5 \((^{11})\). From them, as corollaries, we obtained theorems on the \(\pi\)-solvability of groups for which \(r=1\), and on the \(\pi\)-separability of groups with \(r=2,3\) \((^{7})\).

In the present work a simpler method is given for proving theorems on the solvability of \(\pi d\)-groups for which \(r=t-1\), than the method of obtaining the analogous result in papers \((^{7,8})\), and Theorem 1 and Corollaries 1 and 2 \((^{5})\) are also strengthened. With the help of the theorems of § 2 a new criterion for the speciality of \(\pi d\)-groups is obtained, following from Theorems 3 and 4.

§ 2. We give proofs of the theorems mentioned.

Theorem 1. If a \(\pi d\)-group \(G\) satisfies the relation \(r=t-1\), then it is solvable.

Proof. Since a special group is always solvable, suppose that \(G\) is a nonspecial group. Obviously, among all its Sylow subgroups \(P_1, P_2,\ldots,P_t\) there is at least one attainable. In the sequel, unattainable Sylow \(\pi d\)-subgroups of distinct orders will everywhere be denoted by the first \(s\) indices, and attainable ones by the remaining \(s+1, s+2,\ldots,t\) indices.

We consider separately each of the possible cases:

\[ 1)\quad s=0,\qquad 2)\quad 0<s<r,\qquad 3)\quad s=r. \]

1) \(s=0\), i.e. \(P_1,P_2,\ldots,P_{r+1}\) are attainable in \(G\).

By Lemma 1 \((^{4})\) all of them are invariant in \(G\). In view of the nonspeciality of \(G\), at least one \(q\)-subgroup, for example \(Q_1\), is noninvariant in it. Form the collection of \(\pi d\)-subgroups:

\[ P_1Q_1,\; P_2Q_1,\ldots,\; P_{r+1}Q_1, \tag{*} \]

among which one, for definiteness \(P_{r+1}Q_1\), will necessarily be attainable and invariant in \(G\) (Lemma 3 \((^{11})\) and Lemma 1 \((^{4})\)). Obviously, the \(\pi d\)-subgroups
\(P_rP_{r+1}Q_1,\ldots,\)
\(\ldots, P_1\cdots P_rP_{r+1}Q_1,\)
\(P_1\cdots P_rP_{r+1}Q_1Q_2,\ldots,\)
\(P_1\cdots P_rP_{r+1}Q_1Q_2\cdots Q_{\nu-1}\),
will also be attainable and, consequently, invariant; these form the normal series

\[ G \supset P_1\cdots P_rP_{r+1}Q_1Q_2\cdots Q_{\nu-1} \supset \cdots \supset P_1\cdots P_rP_{r+1}Q \supset \cdots \supset P_rP_{r+1}Q_1 \supset P_{r+1}Q_1 \supset P_{r+1} \supset E, \]

i.e. \(G\) is solvable.

2) \(0<s<r\), i.e. \(P_1,P_2,\ldots,P_s\) are unattainable, while \(P_{s+1},P_{s+2},\ldots,P_{r+1}\) are attainable in \(G\). Then among the \(r+1-s>1\) \(\pi d\)-subgroups of the newly formed collection

\[ P_{s+1}P_1,\; P_{s+2}P_1,\ldots,\; P_{r+1}P \tag{**} \]

only one subgroup is attainable in \(G\). Moreover, it is obvious that \(s<1\). Since the case \(s=0\) has been dealt with, put \(s=1\). Let \(P_2P_1\) be attainable in \(G\). In view of the presence in \(G\) of all \(r\) classes of unattainable isoordinal \(\pi d\)-subgroups, the products
\(P_3P_2P_1,\ldots,P_{r+1}\cdots P_3P_2P_1,\)
\(P_{r+1}\cdots P_3P_2P_1Q_1,\)
\(P_{r+1}\cdots P_3P_2P_1Q_1Q_2,\ldots,\)
\(P_{r+1}\cdots P_3P_2P_1Q_1Q_2\cdots Q_{\nu-1}\)
form a normal series of the required kind, i.e. \(G\) is soluble.

3) \(s=r\), i.e. \(P_1,P_2,\ldots,P_s\) are unattainable, while \(P_{s+1}\) is attainable in \(G\) (here \(r+1-s=1\)).

Obviously, \(P_{s+1}\) is invariant in \(G\). Since \(G\) already has all classes of unattainable isoordinal \(\pi d\)-subgroups, the products
\(P_sP_{s+1},\ldots,P_2\cdots P_sP_{s+1},\)
\(P_1P_2\cdots P_sP_{s+1},\)
\(P_1P_2\cdots P_sP_{s+1}Q_1,\ldots,\)
\(P_1P_2\cdots P_sP_{s+1}Q_1Q_2\cdots Q_{\nu-1}\)
will be attainable and invariant \(\pi d\)-subgroups (by Lemma 1 \((^4)\)), forming a normal series
\(G \supset P_1P_2\cdots P_sP_{s+1}Q_1Q_2\cdots Q_{\nu-1}\supset \cdots \supset P_sP_{s+1}\supset P_{s+1}\supset E\), i.e. \(G\) is soluble.

As follows from the proof, the theorem formulated is also true in the case when all prime divisors of the order of the group belong to the set \(\pi\).

Theorem 2. If a \(\pi d\)-group \(G\) satisfies the relation \(r=t\), then it is soluble.

In proving this theorem, the same cases are considered as in Theorem 1 of the paper \((^5)\); moreover, everywhere instead of \(\pi\)-solubility one obtains solubility, owing to the use of the fundamental theorem on \(\pi\)-separable groups of S. A. Chunikhin \((^{12})\), Lemma 1 \((^4)\), and arguments analogous to those given in \((^{10})\). The theorem is also valid for the case when all prime divisors of the order of the group belong to \(\pi\), i.e. when \(n=1\).

From Theorems 1 and 2 of this note and Theorem 2 \((^5)\) we obtain Corollary 3 \((^5)\), or Theorem 4 \((^7)\), as well as the corollaries on the solubility of \(\pi d\)-groups for which \(r=1\) \((^4)\), and on the solubility of \(\pi d\)-groups with \(r=2\) and \(m\ne p^\alpha\), which improve Theorems 2 and 3 \((^7)\), or Corollaries 1 and 2 \((^5)\).

§ 3. Using Theorems 1 and 2, we now obtain the following results.

Theorem 3. If \(G\) is a nonspecial \(\pi d\)-group and if \(r>1\) and \(n>1\), then \(r\ge t\).

Proof. Suppose the contrary, i.e. suppose that \(r<t\). Since, by Lemma 3 \((^4)\), \(r\ge t-1\), we have \(r=t-1\). If \(s=0\), then, as in case 1) of Theorem 1, the collection \((*)\) would have an attainable and invariant \(\pi d\)-subgroup \(P_{r+1}Q_1\). Then the newly formed \(\pi d\)-subgroup \(P_1P_2Q_1\), obviously, would also be attainable and invariant in \(G\). Since \(r+1\) cannot be equal to 1 or 2, the intersection of \(P_{r+1}Q_1\) with \(P_1P_2Q_1\) would give, by Lemma 3 \((^{11})\), the invariant subgroup \(Q_1\), which is impossible. Hence \(s\ne0\).

We shall show that \(s=1\). Suppose that \(s>1\). Then form the collection of \(\pi d\)-subgroups \((**)\). In it, as in case 2) of Theorem 1, there is only one attainable subgroup \(P_{s+1}P_1\). Since \(G\), by Theorem 1, is soluble, it follows, in view of Hall’s theorem \((^{13})\), that it has a subgroup \(W \supset P_1\) of order \(p_1^{\alpha_1}p_2^{\alpha_2}\). \(W\), obviously, is not conjugate to any of the subgroups of the \(r\) classes of unattainable isoordinal \(\pi d\)-subgroups already existing in \(G\). Therefore it is attainable in \(G\). Since \(s+1\ne2\), \(P_{s+1}P_1\cap W=P_1\), obviously, must be attainable in \(G\), which contradicts the choice of it. Therefore \(s=1\); \(P_1\) is a representative of the unique class of unattainable Sylow \(\pi d\)-subgroups of the group \(G\), while \(P_2,P_3,\ldots,P_{r+1}\) are attainable and, consequently, invariant in \(G\). The \(\pi d\)-subgroup \(P_2P_1\) is also attainable, and in the collection of type \((**)\) one can choose \(r\) classes of unattainable \(\pi d\)-subgroups. Then \(P_3Q_1\) is attainable and invariant (by Lemma 1 \((^4)\)) in \(G\). Form \(P_1P_3Q_1\). Since it is attainable in \(G\), the intersection
\(P_2P_1\cap P_1P_3Q_1=P_1\)
will give an attainable \(\pi d\)-subgroup. Again we have arrived at a contradiction. Hence the relation \(r=t-1\) is impossible. Therefore \(r\ge t\). The theorem is proved.

Corollary. If \(G\) is a nonspecial \(\pi d\)-group and if \(r > 2\), then \(r \ge t\).

Obviously, here two cases are possible: \(n>1\) and \(n=1\). In the first case the corollary is valid by the theorem proved above, and in the second case it is proved analogously.

This theorem can be strengthened for \(r>3\).

Theorem 4. If \(G\) is a nonspecial \(\pi d\)-group and if \(r>3\) and \(n>1\), then \(r \ge t+1\).

Proof. By Theorem 3 the relation \(r<t\) is impossible. Put \(r=t\). We shall first show that \(G\) contains at least one attainable Sylow \(\pi d\)-subgroup.

Suppose all Sylow \(\pi d\)-subgroups of the group \(G\) are unattainable in it. Denote one of them by \(P_1\). Since \(G\), by Theorem 2, is soluble, in view of Hall’s theorem \((^{13})\) it has subgroups \(W_1\) and \(W_2\) of orders respectively \(p_1^{\alpha_1}p_2^{\alpha_2}\) and \(p_1^{\alpha_1}p_3^{\alpha_3}\). Because \(G\) has all \(r\) classes of unattainable \(\pi d\)-subgroups, \(W_1\) and \(W_2\) are attainable, which leads to the attainability of \(P_1\). But this contradicts the supposition. Hence \(G\) has at least one Sylow attainable \(\pi d\)-subgroup. Further, as in Theorem 3, it could be proved that \(s\ne 0\). We now show that \(s=1\). Suppose \(G\) has \(s>1\) classes of unattainable \(\pi d\)-subgroups and let \(P_1\) and \(P_2\) be representatives of two such classes. Then form the set of \(\pi d\)-subgroups

\[ P_{s+1}P_1,\quad P_{s+2}P_1,\ldots,\quad P_rP_1. \tag{***} \]

Consider two possibilities:

1) Suppose all \(\pi d\)-subgroups \((***)\) are unattainable in \(G\). Then, obviously, the newly formed subgroup \(P_{s+1}P_2\) is already attainable in \(G\). For the same reason, the \(\pi d\)-subgroup \(W\) of order \(p_1^{\alpha_1}p_2^{\alpha_2}\), which exists in the soluble group \(G\) by Hall’s theorem \((^{13})\), is also attainable in \(G\). By Lemma 3 \((^{11})\) the intersection
\[ P_{s+1}P_2\cap W=P_2 \]
is attainable in \(G\), which is impossible. Hence in this case \(s=1\).

2) Suppose now that among the \(\pi d\)-subgroups \((***)\) there are some attainable in \(G\). Then one may, obviously, take \(P_{s+1}P_1\) as such a subgroup. Since in \(G\) there are already \(r-1\) classes of unattainable isoordinal \(\pi d\)-subgroups, the newly formed set
\[ P_{s+1}P_2,\quad P_{s+2}P_2,\ldots,\quad P_rP_2 \]
will obviously consist of one attainable and one unattainable \(\pi d\)-subgroup. Suppose, for definiteness, that \(P_{s+1}P_2\) is attainable, and \(P_{s+2}P_2\) is unattainable in \(G\). It is obvious that then
\[ P_{s+1}P_2\cap W=P_2 \]
must be attainable in \(G\), which is impossible. Consequently, also in this second case \(s=1\).

We have shown that \(G\) contains only one class of unattainable Sylow \(\pi d\)-subgroups. Let \(P_1\) be a representative of this class. Form the set of \(\pi d\)-subgroups
\[ P_1P_2,\quad P_1P_3,\ldots,\quad P_1P_r. \]
They are either all unattainable in \(G\), or, by Lemma 3 \((^{11})\), only one of them is attainable in \(G\).

The first case leads to the presence in \(G\) of all \(r\) classes of unattainable \(\pi d\)-subgroups and, consequently, to the attainability in \(G\) of the \(\pi d\)-subgroups:
\[ (P_kP_u)P_1,\quad (P_kP_v)P_1,\quad (P_uP_v)P_1, \]
where \(k,u,v\) are distinct natural numbers, different from 1 and less than \(r\). Hence it is obvious that \(P_1\) is attainable in \(G\). We have arrived at a contradiction.

In the second case, for definiteness, one may suppose that \(P_1P_2\) is attainable in \(G\). Then \(G\) will already contain \(r-1\) classes of unattainable \(\pi d\)-subgroups. Multiplication of the attainable, and hence invariant, subgroup \(P_3P_4\) by the unattainable \(P_1\) will obviously give us the \(\pi d\)-subgroup \(P_1P_3P_4\). If it is attainable in \(G\), then the intersection
\[ P_1P_2\cap P_1P_3P_4=P_1 \]
will be an attainable subgroup in \(G\). We have arrived at a contradiction. If, however, \(P_1P_3P_4\) is unattainable in \(G\), then, in view of the presence in \(G\) of all \(r\) classes of unattainable \(\pi d\)-subgroups, the newly formed \(\pi d\)-subgroup \(P_1(P_3Q)\) is attainable in \(G\). Then the intersection
\[ P_1P_2\cap P_1(P_3Q)=P_1 \]
by Lemma 3 \((^{11})\) will be an attainable subgroup in \(G\).

We have obtained a contradiction. Thus the relation \(r=t\) is impossible. Therefore \(r \geqslant t+1\). The theorem is proved.

Corollary. If \(G\) is a nonspecial \(\pi d\)-group and if \(r>4\), then \(r \geqslant t+1\).

The proof is analogous to the proof of Corollary 1.

Finally, we arrive at the following criterion:

\(\pi d\)-groups for which \(r>3\) when \(n>1\) (or \(r>4\) for any \(n\)) and \(r<t+1\) are special.

Gomel State
Pedagogical Institute named after V. P. Chkalov

Received
11 XII 1962

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