MATHEMATICS

S. A. CHUNIKHIN

ON THE PERMUTABILITY OF FACTORS IN \(\Pi\)-FACTORIZATIONS OF FINITE GROUPS

(Presented by Academician I. M. Vinogradov, 7 XII 1957)

§ 1. In the papers \((^{1,2})\) we introduced the so-called \(\Pi\)-factorization of finite groups, and it was shown that every finite group admits factorizations of this kind in a definite way. In the present paper it is established that among the \(\Pi\)-factorizations of finite groups there always exist \(\Pi\)-factorizations with pairwise permutable factors. Thus Theorem 2 of \((^{1,2})\) receives its natural strengthening.

§ 2. In the present article we shall use the definitions and notation introduced in \((^{1,2})\). In particular, \(\mathfrak G\) denotes a certain finite group of order \(g\). If \(n\), \(n_1\), and \(n/n_1\) are natural numbers and \((n_1,n/n_1)=1\), then \(n_1\) will be called a Sylow divisor of the number \(n\).

Lemma 1. Let \(h\) be a certain reduced \(\Pi\)-integral divisor of the order \(g\) of the group \(\mathfrak G\), and let \(\mathfrak A\) be a certain subgroup of order \(a\) of the group \(\mathfrak G\). Then \(b=(a,h)\) will be a certain reduced \(\Pi\)-integral divisor of \(a\).

Lemma 2. If \(m=h_1h_2\ldots h_t\) is the greatest \(\Pi\)-divisor of the number \(g\), and \(h_1,h_2,\ldots,h_t\) are reduced \(\Pi\)-integral divisors of \(g\), then \(h_1,h_2,\ldots,h_t\) are pairwise relatively prime.

Lemma 3. Let \(m\) and \(h_1,h_2,\ldots,h_t\) be the same as in Lemma 2. If \(\alpha_i=(a,h_i)\), \(i=1,2,\ldots,t\), then \(\alpha=\alpha_1\alpha_2\ldots\alpha_t\) will be the greatest \(\Pi\)-divisor of the order \(a\) of the subgroup \(\mathfrak A\), and the numbers \(\alpha_1,\alpha_2,\ldots,\alpha_t\) will be pairwise relatively prime reduced \(\Pi\)-integral divisors of \(a\).

Lemma 4. If subgroups \(\mathfrak A\) and \(\mathfrak B\) of the group \(\mathfrak G\) are permutable and the greatest \(\Pi\)-divisors of their orders divide the Sylow divisor \(v\) of the number \(g\), then the greatest \(\Pi\)-divisor of the order of the subgroup \(\mathfrak A\mathfrak B\) also divides \(v\).

Lemma 5. Let \(\mathfrak G=\mathfrak A_1\mathfrak A_2\ldots\mathfrak A_r\), where \(\mathfrak A_1,\mathfrak A_2,\ldots,\mathfrak A_r\) are pairwise permutable subgroups of respective orders \(a_1,a_2,\ldots,a_r\). Suppose that the greatest \(\Pi\)-divisor \(a'_\rho\) of the order of the subgroup \(\mathfrak A_\rho\) divides the \(\Pi\)-divisor \(s\) of the number \(g\), and moreover \(s\) is relatively prime to each of the numbers \(a_1,a_2,\ldots,a_{\rho-1},a_{\rho+1},\ldots,a_r\). Then \(a'_\rho=s\).

Lemma 6. Let a normal divisor \(\mathfrak S\) of the group \(\mathfrak G\) contain the subgroup \(\mathfrak S_1\), and let all subgroups of \(\mathfrak G\) conjugate to \(\mathfrak S_1\) in \(\mathfrak G\) already be conjugate to \(\mathfrak S_1\) in \(\mathfrak S\). Then \(\mathfrak G=\mathfrak B\mathfrak S\), where \(\mathfrak B\) is the normalizer of the subgroup \(\mathfrak S_1\) in \(\mathfrak G\).

§ 3. Theorem. Let \(m\) be the greatest \(\Pi\)-divisor of the order \(g\) of the group \(\mathfrak G\). Then to every representation of the number \(m\) in the form of a product
\[ m=h_1h_2\ldots h_t, \]
where each of the factors \(h_1,h_2,\ldots,h_t\) is a certain reduced \(\Pi\)-integral divisor of the order \(g\) of the group \(\mathfrak G\), there corresponds a representation of \(\mathfrak G\) in the form of a product
\[ \mathfrak G=\mathfrak H_1\mathfrak H_2\ldots\mathfrak H_t, \]
where \(\mathfrak H_1,\mathfrak H_2,\ldots,\mathfrak H_t\) are certain pairwise permutable subgroups of \(\mathfrak G\) such that the greatest \(\Pi\)-divisors of their orders are respectively the numbers \(h_1,h_2,\ldots,h_t\).

We shall give the main points of the proof of the theorem. Suppose that there exist groups for which the theorem is not fulfilled. Choose among such groups a group \(\mathfrak G\) having the least order \(g\). Then there exists a nonempty set of prime numbers \(\Pi\) and such a repre-

representation of the number \(m\) in the form \(m=h_1h_2\ldots h_t\) required by the theorem, to which the desired factorization of the group \(\mathfrak G\) does not correspond. It follows from this that \(g\) is a \(\Pi\)-composite number and that \(g>1\) and \(t>1\).

Consider a chief series

\[ \mathfrak G=\mathfrak G_0 \supset \mathfrak G_1 \supset \ldots \supset \mathfrak G_\lambda=\mathfrak E \tag{1} \]

of the group \(\mathfrak G\) (\(\mathfrak E\) is the identity subgroup).

Assigning \(\mathfrak E\) to the special groups, we observe that if all terms of the series (1) were special, then \(\mathfrak G\) itself would be special. But then \(\mathfrak G\), obviously, would admit the required factorization. Therefore among the terms of (1) there exist nonspecial groups. Choose among them such a \(\mathfrak G_\mu=\mathfrak M\) whose index \(\mu\) is greatest. Since \(\mathfrak M\), by assumption, is nonspecial, \(\mathfrak M\ne \mathfrak E\), i.e. \(\mu<\lambda\). Thus \(\mathfrak G_{\mu+1}=\mathfrak N\) exists.

Denote the order of \(\mathfrak N\) by \(n\). Then the order of \(\mathfrak M\) will have the form \(wn\). Represent \(n\) in the form \(n=n_1n_2\), where \(n_2\) is the greatest divisor of \(n\) relatively prime to \(w\). Since \(\mathfrak N\) is a special group, \(\mathfrak N=\mathfrak N_1\mathfrak N_2\), where \(\mathfrak N_1\) and \(\mathfrak N_2\) are subgroups of orders \(n_1\) and \(n_2\), respectively. But \((n_1,n_2)=1\), and therefore \(\mathfrak N_2\) will be invariant in \(\mathfrak G\). Since \(\mathfrak N_2\) is a special group and \((wn_1,n_2)=1\), by the Schur–Zassenhaus theorem \(\mathfrak M\) has a subgroup \(\mathfrak M_1\) of order \(wn_1\), and all subgroups of order \(wn_1\) in \(\mathfrak M\) are conjugate to \(\mathfrak M_1\) in \(\mathfrak M\).

1) \(\mathfrak M_1\) is not invariant in \(\mathfrak G\). Applying Lemma 6 to \(\mathfrak M\) and \(\mathfrak M_1\), we obtain \(\mathfrak G=\mathfrak V\mathfrak M\), where \(\mathfrak V\) of order \(v\) is the normalizer of \(\mathfrak M_1\) in \(\mathfrak G\). But \(\mathfrak M=\mathfrak M_1\mathfrak N_2\), whence \(\mathfrak G=\mathfrak V\mathfrak N_2\). By Lemma 3, \(v=v_1v_2\cdots v_t\), where \(v_i=(v,h_i)\), will be the decomposition required by the theorem of the greatest \(\Pi\)-divisor of \(v\). Since \(\mathfrak M_1\) is not invariant in \(\mathfrak G\), \(v<g\), and the theorem holds for \(\mathfrak V\). Consequently, \(\mathfrak V=\mathfrak F_1\mathfrak F_2\ldots \mathfrak F_t\), where the subgroups \(\mathfrak F_1,\mathfrak F_2,\ldots,\mathfrak F_t\) are pairwise permutable, and the greatest \(\Pi\)-divisors of their orders are respectively equal to \(v_1,v_2,\ldots,v_t\).

Putting \(\nu_i=(n_2,h_i)\), we verify on the basis of Lemma 3 that \(\nu_1,\nu_2,\ldots,\nu_t\) will be pairwise relatively prime Sylow divisors of \(n_2\). But \(\mathfrak N_2\) is a special group; hence we conclude that \(\mathfrak N_2=\mathfrak N'_1\mathfrak N'_2\ldots \mathfrak N'_t\), where \(\mathfrak N'_1,\mathfrak N'_2,\ldots,\mathfrak N'_t\) are invariant subgroups of \(\mathfrak G\) with pairwise relatively prime orders, and the greatest \(\Pi\)-divisors of their orders are respectively equal to \(\nu_1,\nu_2,\ldots,\nu_t\). But then \(\mathfrak G=\mathfrak V\mathfrak N_2=\mathfrak H_1\mathfrak H_2\ldots \mathfrak H_t\), where \(\mathfrak H_i=\mathfrak F_i\mathfrak N'_i\). The subgroups \(\mathfrak H_1,\mathfrak H_2,\ldots,\mathfrak H_t\) are, evidently, pairwise permutable, and with the help of Lemmas 2, 4, and 5 it is established that the greatest \(\Pi\)-divisors of their orders are respectively equal to \(h_1,h_2,\ldots,h_t\). Thus, for \(\mathfrak G\) we have obtained a factorization of the desired form. A contradiction is obtained.

2) \(\mathfrak M_1\) is invariant in \(\mathfrak G\). The subgroup \(\mathfrak M=\mathfrak M_1\mathfrak N_2\) is a nonspecial normal divisor of \(\mathfrak G\); the orders of \(\mathfrak M_1\) and \(\mathfrak N_2\) are relatively prime; \(\mathfrak N_2\) is special and invariant in \(\mathfrak G\). Therefore among the Sylow subgroups of \(\mathfrak M_1\) there exists a subgroup \(\mathfrak R\) not invariant in \(\mathfrak G\). Denoting now by \(\mathfrak V\) the normalizer of \(\mathfrak R\) in \(\mathfrak G\), we obtain, by Lemma 6, that \(\mathfrak G=\mathfrak V\mathfrak M=\mathfrak V\mathfrak M_1\mathfrak N_2\).

Using the fact that \(\mathfrak M_1/\mathfrak R\) is an elementary group, and that \(n_2\) is the greatest divisor of \(n\) relatively prime to \(w\), one can further show that the greatest \(\Pi\)-divisor of the order of \(\mathfrak M_1\) will be a divisor of some number \(h_\rho\) from the series \(h_1,h_2,\ldots,h_t\). Since \(\mathfrak R\) is not invariant in \(\mathfrak G\), \(\mathfrak V\ne \mathfrak G\), and the theorem holds for \(\mathfrak V\). Hence, as in (1), we obtain the factorization
\[ \mathfrak G=\mathfrak F_1\mathfrak F_2\ldots \mathfrak F_t\mathfrak M_1\mathfrak N'_2\mathfrak N'_2\ldots \mathfrak N'_t = \mathfrak H_1\mathfrak H_2\ldots \mathfrak H_t, \]
where
\[ \mathfrak H_1=\mathfrak F_1\mathfrak N'_1,\ldots,\mathfrak H_{\rho-1}=\mathfrak F_{\rho-1}\mathfrak N'_{\rho-1}, \mathfrak H_\rho=\mathfrak F_\rho\mathfrak N'_\rho\mathfrak M_1, \mathfrak H_{\rho+1}=\mathfrak F_{\rho+1}\mathfrak N'_{\rho+1},\ldots, \mathfrak H_t=\mathfrak F_t\mathfrak N'_t. \]
Applying again Lemmas 2, 4, and 5, we establish that the greatest \(\Pi\)-divisors of the orders of \(\mathfrak H_1,\mathfrak H_2,\ldots,\mathfrak H_t\) are respectively equal to \(h_1,h_2,\ldots,h_t\). We have obtained a contradiction.

The contradictions obtained prove the theorem.

Received
2 XII 1957

CITED LITERATURE

1. S. A. Chunikhin, DAN, 108, No. 3, 397 (1956).
2. S. A. Chunikhin. Matem. sborn., 43 (85), No. 1 (1957).