L. A. SHEMETKOV

A NEW \(D\)-THEOREM IN THE THEORY OF FINITE GROUPS

(Presented by Academician I. M. Vinogradov, 1 VII 1964)

§ 1. In this note we use the same notation as in \((^{1})\): \(G\) is a finite group of order \((G)\) (all groups considered are finite); \(\Pi\) is some set of primes; \(m_{\Pi}\) is the greatest \(\Pi\)-divisor \((^{2})\) of the natural number \(m\); \(S_{\Pi}\) is a subgroup of the group \(G\) of order \((G)_{\Pi}\). Properties of the group \(G\): \(E_{\Pi}\)—\(G\) has an \(S_{\Pi}\)-subgroup; \(E_{\Pi}^{\theta}\)—\(G\) has an \(S_{\Pi}\)-subgroup possessing the property \(\theta\); \(D_{\Pi}\)—\(G\) has an \(S_{\Pi}\)-subgroup \(H\), and for any \(\Pi\)-subgroup \(A\) of \(G\) there is an element \(x \in G\) such that
\[ x^{-1}Ax=A^{x}\leq H; \]
\(D_{\Pi}^{\theta}\)—\(G\) has the properties \(E_{\Pi}^{\theta}\) and \(D_{\Pi}\).

Notation for group-theoretic properties: \(n\)—nilpotency, \(s\)—solvability, \(ss\)—supersolvability, \(d\)—dispersiveness in the sense of Ore (the group \(G\) is dispersive in the sense of Ore \((^{10})\) if and only if in \(G\) and in every one of its factor groups there is an invariant Sylow subgroup corresponding to the greatest prime divisor of the order), \(z\)—the property of cyclicity of all Sylow subgroups of the group. \(\Pi(m)\) denotes the set of all distinct prime divisors of the natural number \(m\).

§ 2. In the present note the following will be proved.

Theorem A. If \(K\) is such a normal divisor of the group \(G\) that \(K\) has the property \(E_{\Pi}^{z}\), and \(G/K\) has the property \(D_{\Pi}^{d}\), then \(G\) has the property \(D_{\Pi}^{s}\).

Two consequences can be obtained from this theorem.

Corollary 1. If \(K\) is such a normal divisor of the group \(G\) that \(K\) has the property \(E_{\Pi}^{z}\), and \(G/K\) has the property \(D_{\Pi}^{ss}\), then \(G\) has the property \(D_{\Pi}^{ss}\).

Corollary 2. If every factor of some series of normal divisors of the group \(G\) has the property \(E_{\Pi}^{z}\), then \(G\) has the property \(D_{\Pi}^{ss}\).

§ 3. Lemma 1. Let \(G\) have the property \(E_{\Pi}^{z}\), and let \(M\) be any \(\Pi\)-subgroup of \(G\). Then the normalizer \(N\) of the subgroup \(M\) in \(G\) has the property \(E_{\Pi}^{z}\).

Proof. Suppose that the lemma is true for all groups whose orders are less than \((G)\). By Theorem 16 of \((^{3})\), \(G\) has the property \(D_{\Pi}\), and therefore \(M\) is contained in some \(S_{\Pi}\)-subgroup \(H\) of \(G\). Let \(M\ne 1\). Since \(M\), obviously, is supersolvable, it contains an invariant Sylow subgroup \(P\ne 1\). It is clear that \(N\leq N_{1}\), where \(N_{1}\) is the normalizer of \(P\) in \(G\). If \(P\) is invariant in \(G\), then, by induction in the group \(G/P\), the normalizer \(N/P\) of the subgroup \(M/P\) has an \(S_{\Pi}\)-subgroup \(L/P\). The subgroup \(L\) will be the desired one. Let \(N_{1}\ne G\), and let \(Q\) be a Sylow \(q\)-subgroup \((q\in\Pi)\) of \(N_{1}\). Obviously, \(P^{x}Q^{x}\leq H\) for some \(x\in G\). By Theorem 16 of \((^{3})\), \(P\) and \(P^{x}\) are conjugate in \(H\), and hence the order of the normalizer of the subgroup \(P\) in \(H\) is divisible by \((Q)\). Hence, in view of the arbitrariness of \(q\), it follows that \(N_{1}\) has the property \(E_{\Pi}^{z}\). Since \(M\leq N\leq N_{1}\) and \(N_{1}\ne G\), it follows by induction that \(N\) has the property \(E_{\Pi}^{z}\). The lemma is proved.

Lemma 2. Suppose that Theorem A is valid in the case when \([G/K]\) is divisible by no more than one prime number from \(\Pi\). Let \(G=PK\), \(K\) invariant in \(G\), \(P\cap K=1\), \(P\) is a \(p\)-group, where \(p\) is the greatest

prime number from \(\Pi\) dividing \([G]\), and suppose that \(K\) has property \(E_{\Pi^z}\). Then the normalizer \(N\) of the subgroup \(P\) in \(G\) has property \(E_{\Pi}\).

Proof. Let \(H\) be an \(S_{\pi}\)-subgroup of \(G\). By Lemma 1 from (1), \(H \cap K = A\) is an \(S_{\Pi}\)-subgroup of \(K\). Suppose that \(P\) is nontrivial and is contained in a Sylow \(p\)-subgroup \(H_p\) of \(H\). The lemma is obvious if \((A)\) is a power of a prime number. Therefore suppose that \((A)\) is divisible by at least two distinct prime numbers.

We shall carry out the proof by induction on \((G)\) and on \((G)_{\Pi}\).

Suppose \(H\) has a nilpotent \(S_{p,q}\)-subgroup \(B\), where \(q \in \Pi((H))\), \(q \ne p\). By induction, \(N\) has an \(S_{\Pi q}\)-subgroup \(Q\). Since \(G\) has property \(D_{\Pi}\), we may assume that \(PQ \subseteq H\). From this, taking into account the nilpotency of \(B\), we obtain that the normalizer of the subgroup \(P\) in \(H\) contains \(Q\) and a Sylow \(q\)-subgroup of \(H\), and therefore is an \(S^{\Pi}\)-subgroup of \(N\). Hence let \(H\) satisfy the condition

I. The \(S_{p,q}\)-subgroups of \(H\) are nonnilpotent for every element \(q \in \Pi((H))\) distinct from \(p\).

Let \(K\) be \(p\)-solvable \((^{4,6})\). Then, obviously, \(G\) will be a \(p\)-supersolvable group, i.e., it has a chief series each of whose indices is either equal to \(p\) or is not divisible by \(p\). Let \(\omega\) be all those elements of \(\Pi\) with respect to which \(G\) is supersolvable, and let \(R\) be the maximal invariant \(\omega'\)-subgroup of the group \(G\), where \(\omega'=\Pi((G))\setminus \omega\). Taking into account Theorem XVII from \((^{4})\), we see that the commutator subgroup of \(G/R\) is a \(\Pi\)-group. Hence \(G/R\) is solvable.

Suppose now that the order \(R\) is divisible by some number \(q \in \Pi\), \(q \notin \omega\). If \((R)_q=(G)_q\), then consider the intersection \(R \cap B\), where \(B\) is an \(S_{p,q}\)-subgroup of \(H\). Since \(p \in \omega\), we have \(q \ne p\). Hence \(R \cap B\) is a Sylow \(q\)-subgroup of \(B\), invariant in it. Since by hypothesis \(p>q\), \(B\) will also have an invariant Sylow \(p\)-subgroup, i.e. \(B\) is nilpotent, contrary to I. If, however, \((R)_q<(G)_q\), then, by Theorem 4.2 from \((^{5})\), the group \(G\) is \(q\)-supersolvable, which is impossible in view of \(q \notin \omega\).

Thus \(G/R\) is a solvable group of order divisible by \((G)_{\Pi}\). Hence the subgroup \(N\) is \(\Pi\)-solvable and, by Theorem 2 from \((^{6})\), has property \(E_{\Pi}\). In view of this, suppose that \(K\) satisfies the condition.

II. \(K\) is not \(p\)-solvable.

Let \(A_p\) be a Sylow \(p\)-subgroup of \(A\). In view of II, \((A_p)\ne 1\). Taking into account the condition of the lemma, we see that \(A_p\) is invariant in \(H\). Let \(P_1\) be a subgroup of order \(p\) in \(A_p\). Since \(A_p\) is cyclic, \(P_1\) is invariant in \(H\) and is the unique subgroup of order \(p\) in \(A\). Let \(P_0\) be the normalizer of \(P\) in \(H_p\). It is clear that \(P \ne P_0\). By Dedekind’s identity we have \(P[P_0 \cap K]=P_0\). Obviously, \(P_0 \cap K \subseteq A\) and \((P_0 \cap K)>1\). Since \(P_1\) is the unique subgroup of order \(p\) in \(A\), we have \(P_1 \subseteq P_0 \cap K \subseteq P_0 \subseteq N\).

Suppose now that \((A)\) is divisible by at least three distinct prime numbers. Then one can form such sets \(\sigma\) and \(\tau\) that
\[ \Pi=\sigma \cup \tau,\quad \sigma \cap \tau=p,\quad \sigma \cap \Pi((A))\ne p,\quad \tau \cap \Pi((A))\ne p. \]
By induction, \(N\) has an \(S_{\sigma}\)-subgroup \(U\) and an \(S_{\tau}\)-subgroup \(V\). Obviously, \(P\) is contained in \(U\) and \(V\). Since \(P_1 \subseteq N\), assume, without loss of generality, that \(P_1\) is contained in \(U\) and \(V\). Since \(G\) has property \(D_{\Pi}\), \(U^x \subseteq H\), \(V^y \subseteq H\), where \(x,y\in G\). Therefore \(P_1^x\) and \(P_1^y\) are contained in \(H\), and hence also in \(A\). Since \(P_1\) is the unique subgroup of order \(p\) in \(A\), \(P_1=P_1^x=P_1^y\). Taking this into account, we see that \(P_1\) is invariant in \(U\) and in \(V\), i.e. \(U \subseteq N_1\) and \(V \subseteq N_1\), where \(N_1\) is the normalizer of \(P_1\) in \(G\). Moreover, \(H \subseteq N_1\). By Dedekind’s identity, \(N_1=P[N_1\cap K]\). Since \(A \subseteq N_1\), \(N_1\cap K\) has property \(E_{\Pi^z}\). By induction the normalizer \(N'\) of the subgroup \(P\) in \(N_1\) has property \(E_{\Pi}\) (by Theorem 4.2 from \((^{5})\) and condition II, \(N_1\ne G\)). Since \(U\) and \(V\) are contained in \(N_1\), it follows from this that \(N\) also has property \(N_{\Pi}\).

Finally, let \((A)=p^m q^n\), where \(q\in \Pi\), \(q\ne p\), and let \(S\) and \(T\) be, respectively, a Sylow \(p\)- and a \(q\)-subgroup of \(N\cap K\). We may assume, in view of

property \(D_\Pi\) for \(G\), that \(PT \subseteq H\). Moreover, \(PS \subseteq H^x\) for some \(x \in G\), and hence \(S \subseteq A_p^x\), where \(A_p\) is a Sylow \(p\)-subgroup of \(A\). Since \(H_p^x = P^x A_p^x = P A_p^x\), \(S\) lies in the center of the subgroup \(H_p^x\). Consequently, \(S^{x^{-1}}\) lies in the center of the subgroup \(H_p\), i.e. \(S^{x^{-1}} \subseteq N\). Moreover, \(S^{x^{-1}} \subseteq A_p\). Taking into account that \(A_p\) is a cyclic invariant subgroup of \(A\), we see that \(S^{x^{-1}}T\) is an \(S_\Pi\)-subgroup of \(N \cap K\). Hence, \(PS^{x^{-1}}T\) is an \(S_\Pi\)-subgroup of \(N\), since \(N = P[N \cap K]\).

The lemma is proved.

§ 4. We shall prove Theorem A by induction on \((G)\) and \((G)_\Pi\), according to the scheme proposed by P. Hall for theorems of this type (see (¹), D5).

By a theorem of S. A. Rusakov (³), \(K\) has the property \(D_\Pi\). Hence, by Theorem 1 of S. A. Chunikhin (⁷) (see also (¹)), \(G\) has a solvable \(S_\Pi\)-subgroup \(H\), and all \(S_\Pi\)-subgroups in \(G\) are conjugate. Therefore it remains for us to prove that every maximal \(\Pi\)-subgroup of the group \(G\) is its \(S_\Pi\)-subgroup. Let \(L\) be an arbitrary maximal \(\Pi\)-subgroup of \(G\). By Lemma 1 of (¹), \(HK/K\) is an \(S_\Pi\)-subgroup of \(G/K\), and \(H \cap K\) is an \(S_\Pi\)-subgroup of \(K\). According to the hypothesis, \(HK/K\) is dispersed by Ore, and all Sylow subgroups of \(H \cap K\) are cyclic. Since \(G/K\) has the property \(D_\Pi\), we have \(L \subseteq KH^x\) for some \(x \in G\). If \(KH^x \ne G\), then, by induction, \(KH^x\) has the property \(D_\Pi\), and hence \((L) = (H)\). Therefore

I. \(KH = G\).

Since, according to I, \(G = KLH\), we have
\[ (G)=\frac{(KL)(H)}{(KL\cap H)}, \]
whence \((KL:KL\cap H)=(G:H)\). Therefore, if \(KL \ne G\), then, by induction, \(L^x \subseteq KL\cap H\) for some \(x \in G\). Hence \(L^x = H\). Therefore

II. \(KL = G\).

Let \(\omega = \Pi((L)) \ne \Pi((H))\). In view of Theorem D3 of (¹), we have: \(K\) has the property \(E_{\omega}^{z}\), \(G/K\) has the property \(D_{\omega}^{d}\). Since \((G)_\omega < (G)_\Pi\), by induction \(G\) has the property \(D_\omega\), and hence \(L \subseteq H_\omega^x \subseteq H^x\), where \(H_\omega\) is an \(S_\omega\)-subgroup of \(H\), \(x \in G\). Hence, in view of the maximality of the subgroup \(L\), it follows that \(L = H^x\). Therefore we shall assume that the equality holds:

III. \(\Pi((L))=\Pi((H))\).

Let \(G\) have a minimal invariant \(\Pi\)-subgroup \(X \ne 1\). If \(X \cap K = 1\), then \(KX/X\), isomorphic to \(K\), is invariant in \(G/X\), and moreover
\[ G/X\,/\,KX/X \simeq G/K\,/\,KX/K. \]
Thus, \(G/X\) and \(KX/X\) satisfy the hypothesis of the theorem and, consequently, \(G/X\) has, by induction, the property \(D_\Pi\). Then, by Lemma 4 of (¹), \(G\) has the property \(D_\Pi\). If, however, \(X \cap K \ne 1\), then \(X \subseteq K\). Considering \(G/X\) and \(K/X\), with induction and Lemma 4 of (¹) taken into account, we again arrive at the conclusion that \(G\) has the property \(D_\Pi\). Therefore

IV. \(G\) has no invariant \(\Pi\)-subgroups distinct from the identity.

Suppose now that \(L \cap K = M \ne 1\). Since all Sylow subgroups of \(M\) are cyclic, \(M\) has an invariant Sylow \(p\)-subgroup \(M_p \ne 1\), \(p \in \Pi\). Let \(\overline{N}\) be the normalizer of \(M_p\) in \(G\). Obviously, \(L \subseteq \overline{N}\). By Lemma 1 the subgroup \(N=\overline{N}\cap K\) has the property \(E_\Pi^z\). Moreover, in view of II,
\[ \overline{N}/N \simeq K\overline{N}/K = G/K. \]
Since, in view of IV, \(\overline{N} \ne G\), by induction \(\overline{N}\) has the property \(D_\Pi\), and therefore \((L)=(\overline{N})_\Pi\). By Lemma 1 of (¹), in view of the equality
\[ M=[L\cap \overline{N}]\cap K=L\cap N, \]
the subgroup \(M\) is an \(S_\Pi\)-subgroup of \(N\). Hence \((M_p)=(N)_p\). From this, by virtue of the normalizer property of a \(p\)-group, it follows that \((M_p)=(K)_p\). Since \(K\) has the property \(D_\Pi\), \(M\) is contained in some \(S_\Pi\)-subgroup \(S\) of \(K\). In view of Lemma 1 of (¹), suppose that \(M_p \subseteq M \subseteq S = H \cap K\). Hence, taking Frattini’s lemma into account (see (⁸) or (⁹), Lemma 1), we obtain:
\[ [\overline{N}\cap H]S=H. \]
Thus,
\[ \frac{(\overline{N}\cap H)(S)}{(\overline{N}\cap S)}=(H). \]
But \(\overline{N}\cap S=N\cap S=M\), since \(M\) is an \(S_\Pi\)-subgroup

group from \(N\). Therefore \((\overline N \cap H)=\dfrac{(H)}{S}\cdot (M)\). But from II it follows that \((L)=(G/K)_\Pi(M)\). Consequently, \((\overline N \cap H)=(L)\). Recalling that \(\overline N\) has the property \(D_\Pi\), we obtain \([\overline N\cap H]^x=L\), where \(x\in G\). Hence, \(L\subseteq H^x\), as was required.

Let \(L\cap K=1\). Taking into account Sylow’s theorem, we are convinced of the validity of the following assertion:

V. Theorem A is true if \((G/K)\) is divisible by no more than one prime from \(\Pi\).

We shall also assume that \(K\ne G\), since otherwise the theorem is true.

Let \(p\) be the greatest prime divisor of the number \((G)_\Pi\ne 1\). In view of the condition of the theorem and of remarks II and III, the group \(G/K\) has an invariant Sylow subgroup \(T/K\) of order \(p^m\), \(m>0\). Let \(P=L\cap T\). Since \(G=KL=LT\) and \(L\cap K=1\), we have \((P)=p^m\), i.e., \(P\) is a Sylow subgroup of \(L\), invariant in it. Let \(\overline Q\) be the normalizer of \(P\) in \(G\). Obviously, \(L\subseteq \overline Q\). Since \(T=PK\), \(P\cap K=1\), assertion V allows us to apply Lemma 2, according to which \(\overline Q\cap T\) has the property \(E_\Pi\). Hence, in view of Lemma 1 from \(\left({}^{1}\right)\), \(Q=\overline Q\cap K\) has the property \(E_{\Pi^z}\). Since, by Dedekind’s relation, \(\overline Q=L[\overline Q\cap K]=LQ\), we have \(\overline Q/Q\simeq L\simeq LK/K=G/K\). According to IV, \(\overline Q\ne G\). Consequently, by induction \(\overline Q\) has the property \(D_\Pi\), and hence \(L\) is an \(S_\Pi\)-subgroup of the group \(\overline Q\). From this, in view of the isomorphism \(\overline Q/Q\simeq L\), it follows that \((Q)_\Pi=1\). But then \(P\) will be a Sylow \(p\)-subgroup in \(T\), and hence also in \([H\cap T]^x\). It is not difficult to note, by virtue of the property \(E_{\Pi^z}\) for \(K\), that an \(S_\Pi\)-subgroup \(H\cap T\) of the group \(T\) is supersolvable. Since \(p\) is the greatest prime divisor of \((H)\), it follows that \(P\) is invariant in \([H\cap T]^x\), and hence also in \(H^x\). Thus, \(H^x\) and \(L\) are contained in \(\overline Q\), which has the property \(D_\Pi\). Consequently, \((L)=(H)\).

The theorem is completely proved.

§ 5. Definition. The group \(G\) is called \(\pi\theta\)-closed if it has an invariant \(S_\Pi\)-subgroup possessing the property \(\theta\).

Using the constructions of the proofs of two theorems—theorem A and theorem D5 of Ph. Hall \(\left({}^{1}\right)\)—one can obtain the following, more general result.

Theorem B. Let \(\Pi\) consist of the nonintersecting subsets \(\sigma\) and \(\tau\). Let some normal divisor \(K\) of the group \(G\) have an \(S_\Pi\)-subgroup that is \(\sigma n\)-closed and \(\tau z\)-closed, and let \(G/K\) have the properties \(D_{\Pi^s}\) and \(E_{\tau^d}\). Then \(G\) has the property \(D_{\Pi^s}\).

If \(\sigma\) is empty, then theorem A is obtained from theorem B. If, however, \(\tau\) is empty, then from theorem B, as a special case, one obtains theorem D5 of the paper of Ph. Hall \(\left({}^{1}\right)\).

In the proof of theorem B one uses theorem 16 of the paper of S. A. Rusakov \(\left({}^{3}\right)\) and the theorem of H. Wielandt \(\left({}^{11}\right)\). The following also turns out to be useful.

Lemma 3. Let \(G\) have a normal divisor \(K\), possessing the property \(E_{\Pi^n}\), and let \(M\) be an arbitrary solvable \(\Pi\)-subgroup of the group \(G\). Then \(N\cap K\) has the property \(E_{\Pi^n}\), where \(N\) is the normalizer of the subgroup \(M\) in \(G\).

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

Received
3 VI 1964

CITED LITERATURE

\({}^{1}\) P. Hall, Proc. London Math. Soc., (3), 6, No. 22, 286 (1956).
\({}^{2}\) S. A. Chunikhin, Matem. sborn., 43(85), No. 1, 49 (1957).
\({}^{3}\) S. A. Rusakov, Sibirsk. matem. zhurn., 4, No. 2, 325 (1963).
\({}^{4}\) S. A. Chunikhin, Matem. sborn., 25(67), No. 3, 321 (1949).
\({}^{5}\) H. Wielandt, Abh. Math. Seminar Hamburg, 22, 215 (1958).
\({}^{6}\) S. A. Chunikhin, DAN, 73, No. 1, 29 (1950).
\({}^{7}\) S. A. Chunikhin, DAN 66, No. 2, 165 (1949).
\({}^{8}\) R. Baer, Illinois J. Math., 1, No. 2, 115 (1957).
\({}^{9}\) S. A. Chunikhin, Matem. sborn., 55(97), No. 2, 101 (1961).
\({}^{10}\) O. Ore, Duke Math. J., 5, 431 (1939).
\({}^{11}\) H. Wielandt, Math. Zs., 71, 461 (1959).