UDC 519.44

MATHEMATICS

Academician of the Academy of Sciences of the BSSR S. A. CHUNIKHIN

ON NILPOTENT AND SUPERSOLVABLE SUBGROUPS OF FINITE GROUPS

§ 1. A finite group all of whose Sylow subgroups are invariant is called, as is known, special or nilpotent. In 1929, in the paper (¹), we obtained the following theorems:

I. If for every primary divisor (definition below) \(\delta > 1\) of the order of the commutator subgroup of a finite group of order \(g\) the condition \((\delta - 1, g) = 1\) is satisfied, then the group is special.

II. If for every primary divisor \(\delta > 1\) of a natural number \(g\) the condition \((\delta - 1, g) = 1\) is satisfied, then all groups of order \(g\) are special.

Thirty years later Theorem II was obtained again by G. Pazderski (²), to whom our paper (¹), apparently, remained unknown. The paper (¹) was continued by us in (³, ⁴). The basic arithmetical idea of (¹) and its continuations (³, ⁴), in combination with the main feature of our method of indices (⁵) and factorizations of groups (⁹) (to look for subgroups whose orders are products of the indices of certain series of subgroups), we now apply to the detection of nilpotent and supersolvable subgroups of a finite group. Let us note the simplicity of formulation of Theorem 2 given below.

§ 2. Notation. A primary divisor of a natural number is a divisor that is a power of a prime number (including 1); if \(F\) is a finite sequence of natural numbers, then \(\overline F\), when \(F\) is nonempty, denotes the product of all elements of \(F\), and when \(F\) is empty is equal to 1; \(\mathfrak G\) is a finite group; \(|\mathfrak G|\) is its order; \(k\) is the order of its commutator subgroup; \(\mathfrak E\) is its identity subgroup; an invariant series of \(\mathfrak G\) is a series of subgroups all of whose terms are invariant in \(\mathfrak G\); a refinement of an invariant series \(R\) of the group \(\mathfrak G\) (see (⁵))

\[ R_f:\quad \mathfrak G=\mathfrak G_0 \supseteq \mathfrak G_1 \supseteq \cdots \supseteq \mathfrak G_{\beta-1} \supseteq \mathfrak F_\beta \supseteq \mathfrak G_\beta \supseteq \mathfrak F_{\beta+1} \supseteq \mathfrak G_{\beta+1} \supseteq \cdots \cdots \mathfrak G_{\nu-1} \supseteq \mathfrak F_\nu \supseteq \mathfrak G=\mathfrak E \]

by means of subgroups \(\mathfrak F_i\), \(i=\beta,\ldots,\nu\), we shall call an index series of \(\mathfrak G\) if each \(\mathfrak F_i\) satisfies the “conjugacy condition”: every subgroup of \(\mathfrak G_{i-1}\) conjugate to \(\mathfrak F_i\) in \(\mathfrak F_\beta\) is already conjugate to \(\mathfrak F_i\) in \(\mathfrak G_{i-1}\); the subgroups \(\mathfrak F_i\) will be called the factorial subgroups of the series \(R\) or of the series \(R_f\); if \(|\mathfrak F_i/\mathfrak G_i|=f_i\), then \(f_\beta f_{\beta+1}\cdots f_\nu=h\) will be called the index of the group \(\mathfrak G\) or the index of the series \(R\), and \(\mathfrak F_i/\mathfrak G_i\) its factors; if \(\mathfrak A/\mathfrak B\), then \(\mathfrak A\) will be called the numerator and \(\mathfrak B\) the denominator of the factor group \(\mathfrak A/\mathfrak B\); a subgroup \(\widetilde{\mathfrak F}_i\), \(\beta \le n \le \nu\), of \(\mathfrak F_\beta\) will be called a nilpotent \(d\)-extension (a supersolvable \(d\)-extension) of \(\mathfrak F_i\) in \(\mathfrak F_\beta\), if \(\mathfrak F_i\) is invariant in \(\widetilde{\mathfrak F}_i\) and \(\widetilde{\mathfrak F}_i/\mathfrak F_i\) is a nilpotent group (a supersolvable group) of order \(d\); Schmidt groups are minimal non-nilpotent groups (¹⁰); a maximal subgroup is, when \(\mathfrak G \ne \mathfrak E\), a proper subgroup of \(\mathfrak G\) (i.e. \(\ne \mathfrak G\)) that is not a proper subgroup of any proper subgroup of \(\mathfrak G\) (and when \(\mathfrak G=\mathfrak E\), it is \(\mathfrak E\) itself); \(E(\mathfrak G,n,\mathfrak A,a)\) means that \(\mathfrak G\) has a nilpotent subgroup \(\mathfrak A\) and \(|\mathfrak A|=a\); \(E(\mathfrak G,n,a)\) means that \(\mathfrak G\) has a nilpotent subgroup of order \(a\).

§ 3. Definition 1. Let \(r\) and \(\delta_1,\delta_2,\ldots,\delta_m\) be natural numbers and let \(H_i=\{\delta_1,\delta_2,\ldots,\delta_i\}\), \(1\le i\le m\). Then \(H=H_m\) will be called a sequence of type \(rC\), or an \(rC\)-sequence, if for every \(i=1,\ldots,m\) and every primary divisor \(d_i\) of the number \((\delta_i,r)\) it follows from \(d_i>1\) that \((d_i-1,\overline H_i)=1\). For \(m=1\), an \(rC\)-sequence will be called an \(rC\)-number. In the notation \(rC\)-sequence and \(rC\)-number we shall agree to omit \(r\) when \(r=\delta_1,\delta_2\ldots\delta_m\). The empty sequence will also be assigned to type \(rC\).

Theorem 1. Let \(d\) be any natural number. If the \(kdC\)-sequence (the \(C\)-sequence) \(H\) is a subsequence of the sequence of all indices of some invariant series \(R\) of the group \(\mathfrak G\), then \(\mathfrak G\) has a nilpotent subgroup \(\mathfrak H\) of order \(\overline H\). If \((\overline H,k)=1\), or if \(\overline H\) is free of cubes of primes, then \(\mathfrak H\) is abelian. If \(R\) is a principal series of \(\mathfrak G\), then all terms \(H\) are primary numbers.

Proof. It is obvious that it suffices to prove the theorem for the case when \(H\) is a \(kC\)-sequence and the series \(R\) is principal. By hypothesis, each term \(h\) of \(H\) is a \(kC\)-number. Hence, taking into account that the order of the commutant of the corresponding factor group of the series \(R\) divides \(k\), we conclude, by our theorem I, that this factor group will be a nilpotent group. But, as a factor of a principal series, it is elementary. Therefore its order \(h\) is a primary number. Now let \(\mathfrak G\) be a counterexample of least order to the theorem being proved, and let \(H\) be that \(kC\)-sequence for which \(\mathfrak G\) has no corresponding subgroup. Then \(\overline H>1\), i.e. \(|\mathfrak G|>1\). Let
\[ \mathfrak G=\mathfrak G_0 \supset \mathfrak G_1 \supset \cdots \supset \mathfrak G_\lambda=\mathfrak E \]
be the series \(R\), and let \(|\mathfrak G_{\lambda-1}|=h_\lambda\). Of course, \(h_\lambda>1\).

1) \(h_\lambda\in H\). The theorem is obviously true for \(\mathfrak G/\mathfrak G_{\lambda-1}\) with respect to the \(kC\)-sequence \(H\setminus\{h_\lambda\}\). Therefore \(\mathfrak G/\mathfrak G_{\lambda-1}\) contains a nilpotent subgroup \(\mathfrak N/\mathfrak G_{\lambda-1}\) of order \(\overline H/h_\lambda\). By the preceding, \(h_\lambda=q^\omega\), \(\omega>0\), \(q\) a prime. If \(\overline H/h_\lambda=1\), then \(H=\{h_\lambda\}\), and \(\mathfrak G_{\lambda-1}\) will be the desired nilpotent subgroup. Let \(\overline H/h_\lambda>1\), and let \(p\) be an arbitrary prime divisor of the number \(\overline H/h_\lambda\). Since \(\mathfrak N/\mathfrak G_{\lambda-1}\) is a nilpotent group, its Sylow \(p\)-subgroup \(\mathfrak P/\mathfrak G_{\lambda-1}\) of order \(p^\alpha\), \(\alpha>0\), is invariant in it. Then \(\mathfrak P\) of order \(p^\alpha q^\omega\) is invariant in \(\mathfrak N\). Suppose \(p\ne q\). If a Sylow \(p\)-subgroup from \(\mathfrak P\) is invariant in it, then it will be an invariant Sylow \(p\)-subgroup of \(\mathfrak N\). Suppose a Sylow \(p\)-subgroup from \(\mathfrak P\) is not invariant in it. Then, by theorem 3.2 from \((1)\), it follows that \(\mathfrak P\) contains a Schmidt \(p\)-nilpotent group \(\mathfrak S\) of order \(p^{\alpha_1}q^{\omega_1}\), \(\alpha_1,\omega_1>0\). By lemma 1 from \((1)\), \(q^{\omega_1}\) is the order of the commutant of \(\mathfrak S\), and therefore \(q^{\omega_1}\) divides \(k\). By hypothesis, \(h_\lambda=q^\omega\in H\), and \(p\) divides \(\overline H/h_\lambda\). Hence, taking into account that \(H\) is a \(kC\)-sequence, it follows that \(q^x\not\equiv 1\pmod p\) for all \(1\le x\le \varphi_1\). But then the Sylow \(p\)-subgroup from \(\mathfrak S\) is invariant in it, which is impossible. Suppose \(p=q\). Then \(\mathfrak P\) will be an invariant \(q\)-Sylow subgroup in \(\mathfrak N\). Thus, \(E(\mathfrak G,n,\mathfrak N,\overline H)\). A contradiction.

2) \(h\in H\). The theorem is obviously true for \(\mathfrak G/\mathfrak G_{\lambda-1}\) with respect to the \(kC\)-sequence \(H\). Therefore \(\mathfrak G/\mathfrak G_{\lambda-1}\) contains a nilpotent subgroup \(\mathfrak N/\mathfrak G_{\lambda-1}\) of order \(\overline H>1\). For \(\mathfrak N/\mathfrak G_{\lambda-1}\), one can obviously construct a principal series with a nondecreasing sequence of prime indices \(F\), and moreover \(\overline F=\overline H\).

2.1) There exists a maximal subgroup \(\mathfrak M\) of the group \(\mathfrak N\) not containing \(\mathfrak G_{\lambda-1}\). Then \(\mathfrak N=\mathfrak M\mathfrak G_{\lambda-1}\). In view of \(\mathfrak N/\mathfrak G_{\lambda-1}\simeq \mathfrak M/\mathfrak M\cap\mathfrak G_{\lambda-1}\) and \(|\mathfrak M|<|\mathfrak G|\), the theorem is true for \(\mathfrak M\) with respect to the above-mentioned \(kC\)-sequence \(F\). Therefore \(E(\mathfrak M,n,F)\), \(\overline F=\overline H\). A contradiction.

2.2) \(\mathfrak G_{\lambda-1}\) is contained in every maximal subgroup \(\mathfrak M\) of the group \(\mathfrak N\). Then \(\mathfrak M/\mathfrak G_{\lambda-1}\) will be a maximal subgroup of the group \(\mathfrak N/\mathfrak G_{\lambda-1}\). But \(\mathfrak N/\mathfrak G_{\lambda-1}\) is a nilpotent group, and therefore \(\mathfrak M/\mathfrak G_{\lambda-1}\) is invariant in it. This means that all maximal subgroups \(\mathfrak M\) of \(\mathfrak N\) are invariant in it. Then \((^6)\) the group \(\mathfrak N\) of order \(\overline H h_\lambda\) is a nilpotent group, and therefore \(E(\mathfrak N,n,\overline H)\). A contradiction. The assertions about the abelianness of the subgroup of order \(\overline H\) are obvious.

Theorem 2. If a nondecreasing sequence of prime numbers \(P\) is a subsequence of the sequence of all indices of some principal series of the group \(\mathfrak G\), then \(\mathfrak G\) has a nilpotent subgroup of order \(\overline P\).

Proof. Apply theorem 1 to \(P\).

§ 4. Let
\[ R_i:\ \mathfrak F_i=\mathfrak F_i^{(0)} \supset \mathfrak F_i^{(1)} \supset \cdots \supset \mathfrak F_i^{(\nu_i)}=\mathfrak G_i,\quad i=\beta,\ldots,\nu \]
be a series of invariant subgroups of the factorial subgroup \(\mathfrak F_i\), and let \(\Phi_i\) be some subsequence of the sequence of all factors of the invariant

of a series of subgroups \(R_i\) (\(\Phi_i\) may, in particular, be empty). Let \(a_i\) also be the sequence of the orders of all elements of \(\Phi_i\). The numerators and denominators of all factor groups from \(\Phi_i\) form a subsequence \(A_i\) of the sequence \(R_i\). We shall call the sequence of subgroups \(A_i\) an insertion of type \(a_i\) of the group \(\mathfrak F_i\). The numerator and denominator of each term of \(\Phi_i\) will be called a diad from the insertion \(A_i\). Let
\[ A_i^*=A_\beta\cup A_{\beta+1}\cup\cdots\cup A_i \]
and
\[ a_i^*=a_\beta\cup a_{\beta+1}\cup\cdots\cup a_i. \]
Then \(A_i^*\) will be called an insertion of type \(a_i^*\) (of the series \(R_f\) or of the group \(\mathfrak G\)). We shall regard \(a_i^*\) as given if, for \(j=\beta,\ldots,i\), the insertion \(A_i\) is specified together with its division into diads. For \(i=\nu\) put \(A_\nu^*=A\) and \(a_\nu^*=a\). A diad from any \(A_i\) will also be called a diad from \(A\). For \(i<\nu\) one may regard \(A_i^*\) as such an \(A\) for which \(a_{i+1},a_{i+2},\ldots,a_\nu\) are empty.

Definition 2. The insertion \(A_i^*,\ i=\beta,\ldots,\nu\), will be called an \(r\)-insertion (a supersoluble insertion) if: 1) \(a_i^*\) is an \(rC\)-sequence (a sequence of prime numbers); 2) if \(i>\beta\), then for \(j=\beta+1,\ldots,i\) every subgroup from \(A_j\) is invariant in any nilpotent \(\overline{a}_{j-1}^*\)-extension (supersoluble \(\overline{a}_{j-1}^*\)-extension) of the subgroup \(\mathfrak F_j\) in \(\mathfrak F_\beta\).

Definition 3. The type \(a_i^*\) of a \(kd\)-insertion (supersoluble insertion) \(A_i^*,\ i=\beta,\ldots,\nu\), will be called a nilpotent (supersoluble) sequence of index \(h\) of the group \(\mathfrak G\) (\(d\) natural).

Of interest, because of its simplicity, is the case \(kd=|\mathfrak G|\), which does not require knowledge of the number \(k\).

An example of a nilpotent sequence \(a\) is obtained if, for \(i=\beta,\ldots,\nu\), all subgroups from \(A_i\) are characteristic in \(\mathfrak F_i\), and \(a\) is a nondecreasing sequence of primes. Then \(a\) is a nilpotent sequence. Thus, if all factors of an indexial \(h\) are cyclic and, for every \(i>\beta\), the greatest prime divisor of the number \(f_i\) is not greater than the least prime divisor of the number \(f_{i-1}\), then \(h=\overline a\). Such an indexial will be called cyclically nondecreasing.

§ 5. Theorem 3. If \(a\) is a nilpotent supersoluble sequence of indexial \(h\) of the group \(\mathfrak G\), then \(\mathfrak G\) has a nilpotent supersoluble subgroup \(\mathfrak H\leq \mathfrak F_\beta\) of order \(\overline a\).

Proof. We first prove the theorem for the case of nilpotent sequences. Let \(\mathfrak G\) be a counterexample of least order to the theorem being proved. Then \(\mathfrak G\) has no nilpotent subgroup of order \(\overline a\), where \(a\) is the type of some \(kd\)-insertion \(A\) of some series of the form \(R_f\). This means that \(\overline a>1\), whence also \(|\mathfrak G|>1\). Then in the series \(R\) there exists \(\mathfrak G_\lambda=\mathfrak G\) with the least number \(\lambda\). Therefore \(\mathfrak G_{\lambda-1}\ne\mathfrak G\) and \(\overline a_{\lambda+1}=\overline a_{\lambda+2}=\cdots=\overline a_\nu=1\). For \(\beta=\lambda\) we have \(\overline a=\overline a_\beta\), where \(a_\beta\) is a \(kdC\)-sequence of certain indices of the invariant series \(A_\lambda\) of the group \(\mathfrak F_\lambda\). By Theorem 1, \(E(\mathfrak F_\lambda,n,\overline a)\). A contradiction. Hence \(\beta<\lambda-1\). Let
\[ L:\quad \mathfrak F_\beta \supseteq \cdots \supseteq \mathfrak G_{\lambda-1} \]
be a section of the series \(R_f\), refined by the \(r\)-insertion \(A_{\lambda-1}^*\) of type \(a_{\lambda-1}^*\) (\(r\) arbitrary). Factoring all terms of \(L\) by \(\mathfrak G_{\lambda-1}\), we obtain the series
\[ M:\quad \mathfrak F_\beta/\mathfrak G_{\lambda-1}\supseteq\cdots\supseteq \mathfrak G_{\lambda-1}/\mathfrak G_{\lambda-1}. \]
A direct verification shows that (a) is true: if the series \(L\) has an \(r\)-insertion of type \(a_{\lambda-1}^*\), then \(M\) also has an \(r\)-insertion of the same type.

a) \(|\mathfrak F_\lambda|=1\). Taking account of (a) for \(r=kd\) and of the fact that the order of the commutator subgroup of \(\mathfrak F_\beta/\mathfrak G_{\lambda-1}\) divides \(k\), and
\[ |\mathfrak F_\beta/\mathfrak G_{\lambda-1}|<|\mathfrak G|, \]
we see that the theorem for \(\mathfrak F_\beta/\mathfrak G_{\lambda-1}\) is valid with respect to \(a_{\lambda-1}^*\). Therefore
\[ E(\mathfrak F_\beta/\mathfrak G_{\lambda-1},n,\mathfrak R/\mathfrak G_{\lambda-1},\overline a_{\lambda-1}^*). \]
But, since \(\mathfrak F_\lambda=\mathfrak G\), \(\overline a_\lambda=1\). Hence \(\overline a_{\lambda-1}^*=\overline a\), i.e.
\[ |\mathfrak R/\mathfrak G_{\lambda-1}|=\overline a. \]
Then \(\mathfrak R/\mathfrak G_{\lambda-1}\) has a chief series with a nondecreasing sequence of prime indices, i.e. \(\mathfrak R\) has an invariant series passing through \(\mathfrak G_{\lambda-1}\) and having, on the section from \(\mathfrak R\) to \(\mathfrak G_{\lambda-1}\), a nondecreasing sequence of prime indices whose product is equal to \(\overline a\). Hence, by Theorem 2, we conclude that \(E(\mathfrak R,n,\overline a)\). A contradiction.

b) \(|\mathfrak F_\lambda|>1\) and \(\mathfrak F_\lambda\) is invariant in \(\mathfrak F_\beta\). As in case a), we ascertain the existence of
\[ E(\mathfrak F_\beta/\mathfrak G_{\lambda-1},n,\mathfrak R/\mathfrak G_{\lambda-1},\overline a_{\lambda-1}^*) \]
and of an invariant series
\[ \mathfrak R\supseteq\cdots\supseteq\mathfrak G_{\lambda-1}\supseteq\mathfrak F_\lambda\supseteq\mathfrak G, \]
in which, from \(\mathfrak R\) to \(\mathfrak G_{\lambda-1}\), all indices form a nondecreasing

sequence \(\sigma\) of prime numbers, with \(\bar{\sigma}=\bar a_{\lambda-1}^*\). But then \(\sigma\) will also be the sequence of indices of the invariant series
\[ \mathfrak N/\mathfrak F_\lambda \supseteq \cdots \supseteq \mathfrak G_{\lambda-1}/\mathfrak F_\lambda \supseteq \mathfrak F_\lambda/\mathfrak F_\lambda \]
on the segment from \(\mathfrak N/\mathfrak F_\lambda\) to \(\mathfrak G_{\lambda-1}/\mathfrak F_\lambda\). Hence, by Theorem 2, we conclude that
\[ E(\mathfrak N/\mathfrak F_\lambda,n,\mathfrak N^*/\mathfrak F_\lambda,\bar a_{\lambda-1}^*). \]
Then there exists an invariant series
\[ S:\quad \mathfrak N^*\supseteq \cdots \supseteq \mathfrak F_\lambda\supseteq \mathfrak G \]
with sequence of indices \(\sigma\) on the segment from \(\mathfrak N^*\) to \(\mathfrak F_\lambda\). We refine the series \(S\) on the segment from \(\mathfrak F_\lambda\) to \(\mathfrak G\) by the \(kd\)-insertion \(A_\lambda\) and obtain the series \(S^*\). All subgroups from \(A_\lambda\), according to 2) of Definition 2, will be invariant in \(\mathfrak N^*\). Further, the sequence of those indices of the series \(A_\lambda\) which enter into \(a_\lambda\) will consist, according to Definition 1 and 1) of Definition 2, of the numbers \(\delta_j\) with the following property: every prime divisor \(>1\), \(d_j\), of the number \((\delta_j,kd)\) is such that
\[ (d_j-1,\bar a_{\lambda-1}^*\bar a_{\lambda j})=1, \]
where \(a_{\lambda j}\) denotes the sequence of all members of \(a_\lambda\) which precede \(\delta_j\), and \(\delta_j\) itself. But
\[ \bar a_{\lambda-1}^*=\bar{\sigma}. \]
Then
\[ (d_j-1,\bar{\sigma}\bar a_{\lambda j})=1. \]
This shows that the sequence \(\bar H\), composed of all indices of the series \(S^*\) on the segment from \(\mathfrak N^*\) to \(\mathfrak F_\lambda\) and of all indices of the series \(A_\lambda\) entering into \(a_\lambda\), will be a \(kdC\)-sequence, and moreover
\[ \bar H=\bar{\sigma}\bar a_\lambda=\bar a_{\lambda-1}^*\bar a_\lambda=\bar a. \]
Applying Theorem 1, we see that
\[ E(\mathfrak N^*,n,\bar a). \]
Contradiction.

c) \(|\mathfrak F_\lambda|>1\) and \(\mathfrak F_\lambda\) is not invariant in \(\mathfrak F_\beta\). Let \(\mathfrak B\) be the normalizer of \(\mathfrak F_\lambda\) in \(\mathfrak F_\beta\). Then \((^7)\)
\[ \mathfrak F_\beta=\mathfrak B\mathfrak G_{\lambda-1} \]
and there exists an isomorphism \(\varphi\) of the group \(\mathfrak F_\beta/\mathfrak G_{\lambda-1}\) onto \(\mathfrak B/\mathfrak B_{\lambda-1}\), where
\[ \mathfrak B_{\lambda-1}=\mathfrak B\cap \mathfrak G_{\lambda-1}. \]
Applying \(\varphi\) to the terms of the series \(M\), we obtain the series
\[ N:\quad \mathfrak B/\mathfrak B_{\lambda-1}\supseteq \cdots \supseteq \mathfrak B_\lambda/\mathfrak B_{\lambda-1}\supseteq \mathfrak B/\mathfrak B_{\lambda-1}\supseteq \cdots \supseteq \mathfrak B_{\lambda-1}/\mathfrak B_{\lambda-1}. \]
Taking into account (a) for \(r=kd\), we see that \(M\) has a \(kd\)-insertion of type \(a_{\lambda-1}^*\). But then, obviously, \(N\) also has a \(kd\)-insertion of the same type, and \(k\), obviously, divides the order of the commutator subgroup of \(\mathfrak B/\mathfrak B_{\lambda-1}\). Since \(\mathfrak F_\lambda\) is not invariant in \(\mathfrak F_\beta\), we have
\[ |\mathfrak B|<|\mathfrak G|, \]
and the theorem holds for \(\mathfrak B/\mathfrak B_{\lambda-1}\) with respect to \(a_{\lambda-1}^*\). Therefore
\[ E(\mathfrak B/\mathfrak B_{\lambda-1},n,\mathfrak N/\mathfrak B_{\lambda-1},a_{\lambda-1}^*). \]
Then there exists an invariant series
\[ \mathfrak N\supseteq \cdots \supseteq \mathfrak B_{\lambda-1}\supseteq \mathfrak F_\lambda\supseteq \mathfrak E, \]
whose indices on the segment from \(\mathfrak N\) to \(\mathfrak B_{\lambda-1}\) form a nonincreasing sequence \(\sigma\) of prime numbers. But then \(\sigma\) will also be the sequence of indices of the invariant series
\[ \mathfrak N/\mathfrak F_\lambda\supseteq \cdots \supseteq \mathfrak B_{\lambda-1}/\mathfrak F_\lambda\supseteq \mathfrak F_\lambda/\mathfrak F_\lambda \]
on the segment from \(\mathfrak N/\mathfrak F_\lambda\) to \(\mathfrak B_{\lambda-1}/\mathfrak F_\lambda\). Hence, by Theorem 2, it follows that
\[ E(\mathfrak N/\mathfrak F_\lambda,n,\mathfrak N^*/\mathfrak F_\lambda,\bar a_{\lambda-1}^*). \]
We can now apply to \(\mathfrak N^*/\mathfrak F_\lambda\) the arguments of b), and, taking into account that \(\mathfrak N^*\subseteq \mathfrak F_\beta\), arrive at a contradiction. For the case where \(a\) is a supersolvable sequence, it is necessary to repeat the preceding arguments, replacing throughout nilpotency by supersolvability; a \(kdC\)-sequence by a sequence of prime numbers; \(kdC\)-insertions by supersolvable insertions; a nonincreasing sequence of prime numbers by a sequence of prime numbers; references to Theorem 2 by references to the theorem of L. A. Shemetkov \((^8)\) (corollary).

Theorem 4. If \(h\) is a Chunikhin nonincreasing index of the group \(\mathfrak G\), then \(\mathfrak G\) has a nilpotent subgroup of order \(h\).

Proof. As was shown above, such an index
\[ h=\bar a, \]
where \(a\) is a nilpotent sequence. Consequently, Theorem 3 can be applied to it.

Theorems 3 and 4 are analogues of Theorems 1, 4, and 5 of \((^7)\). For Theorems 2, 3, and 5 of \((^7)\), corresponding analogues can also be indicated.

Gomel Laboratory of the Institute of Mathematics
Academy of Sciences of the BSSR
Gomel State University

Received
6 V 1970

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