M. I. KARGAPOLOV

ON THE FACTORIZATION OF (\Pi)-SEPARABLE GROUPS

(Presented by Academician P. S. Aleksandrov, 16 I 1957)

In the work of S. A. Chunikhin ((^{1})) the following theorem is proved.
Let (m>1) be the greatest (\Pi)-Sylow divisor of the order (g) of a finite (\Pi)-separable group (\mathfrak{G}), and let
[
m=p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k},
]
where (p_1,p_2,\ldots,p_k) are all the distinct prime divisors of the number (m). Then the group (\mathfrak{G}) can be represented in the form of a product
[
\mathfrak{G}=\mathfrak{P}_1\cdot \mathfrak{P}_2\cdots \mathfrak{P}_k, \tag{1}
]
where (\mathfrak{P}_1,\mathfrak{P}_2,\ldots,\mathfrak{P}_k) are certain subgroups of (\mathfrak{G}), having as the greatest (\Pi)-Sylow divisors of their orders, respectively, the numbers
[
p_1^{\alpha_1},\ p_2^{\alpha_2},\ldots,\ p_k^{\alpha_k}.
]

There the question was also posed of the possibility of choosing pairwise permutable factors of the product (1). In the present note this question is answered affirmatively (see Theorem 1), and the theorem of S. A. Chunikhin, thus refined, is extended to a certain class of infinite groups.

Definitions. A subgroup (\mathfrak{H}) of a group (\mathfrak{G}) will be called (\Pi)-closed, or closed relative to the set of prime numbers (\Pi), if every Sylow (p)-subgroup, (p\in\Pi), of the subgroup (\mathfrak{H}) is a Sylow subgroup in the group (\mathfrak{G}). In particular, if (\Pi) coincides with the set (\Pi(\mathfrak{H})) of prime divisors of the orders of the elements of the subgroup (\mathfrak{H}), then the latter will be called closed in the group (\mathfrak{G}).

A collection of pairwise permutable subgroups
[
\mathfrak{P}_1,\ \mathfrak{P}_2,\ldots,\mathfrak{P}_n,\ldots
]
of a locally finite group (\mathfrak{G}) will be called separable relative to the set of prime numbers (\Pi), or a (\Pi)-separable basis of the group (\mathfrak{G}), if the following conditions are satisfied: 1) (\mathfrak{G}={\mathfrak{P}_1,\mathfrak{P}_2,\ldots,\mathfrak{P}_n,\ldots}); 2) the subgroup (\mathfrak{P}_n), (n=1,2,\ldots), is closed relative to the set (\Pi\cap\Pi(\mathfrak{P}_n)); 3) if
[
\Pi\cap\Pi(\mathfrak{G})={p_1,\ p_2,\ldots,p_n,\ldots},
]
then
[
\Pi\cap\Pi(\mathfrak{P}_n)={p_n}
]
and
[
\Pi\cap\Pi(\mathfrak{P}_i)\ne \Pi\cap\Pi(\mathfrak{P}_j)
\quad\text{for } i\ne j.
]

Theorem 1. A finite (\Pi)-separable group possesses at least one (\Pi)-separable basis.

Proof. Suppose that the theorem is not true for all finite (\Pi)-separable groups. Among all finite (\Pi)-separable groups for which the theorem does not hold, choose some (\mathfrak{G}) of least order (g). Denote by
[
\mathfrak{N}=\mathfrak{P}'_1\times \mathfrak{P}'_2\times\cdots\times \mathfrak{P}'_k
]
the maximal special invariant (\Pi)-subgroup of the group (\mathfrak{G}), where (\mathfrak{P}_i), (i=1,2,\ldots,k), is a Sylow (p_i)-subgroup of the group (\mathfrak{N}),
[
p_i\in \Pi\cap\Pi(\mathfrak{G})={p_1,\ p_2,\ldots,p_k},
]
and by (\mathfrak{N}_1/\mathfrak{N}) some minimal normal divisor of the factor group (\mathfrak{G}/\mathfrak{N}).

The following two cases are possible.

1. The order of the factor (\mathfrak{N}_1/\mathfrak{N}) is not divisible by any prime number from the set (\Pi). Then, by a well-known theorem of Schur, in (\mathfrak{N}_1) there exists a subgroup (\mathfrak{A}) of order equal to the order of the factor (\mathfrak{N}_1/\mathfrak{N}), and moreover all sub-

the subgroups of the normal divisor (\mathfrak N_1) of such order are conjugate to one another. Consequently, the index of the normalizer (\mathfrak N(\mathfrak A){\mathfrak N_1}) in the normal divisor (\mathfrak N_1) is equal to the index of the normalizer (\mathfrak N(\mathfrak A)) in the group (\mathfrak G). If the index
(s=[\mathfrak G:\mathfrak N(\mathfrak A){\mathfrak G}]) of the normalizer (\mathfrak N(\mathfrak A)) in the group (\mathfrak G) is not equal to 1, then the order of the normalizer (\mathfrak N(\mathfrak A){\mathfrak G}) is less than (g), and hence, by the induction hypothesis, the group (\mathfrak N(\mathfrak A)) has at least one (\Pi)-separable basis
(\mathfrak P''_1,\mathfrak P''_2,\ldots,\mathfrak P''_k).

Putting (\mathfrak P_i=\mathfrak P'_i\cdot\mathfrak P''_i), (i=1,2,\ldots,k), we obtain a (\Pi)-separable basis of the group (\mathfrak G):
(\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k).

Indeed, the permutability of the subgroups (\mathfrak P_i) and (\mathfrak P_j) follows from the permutability, between themselves, of the subgroups
(\mathfrak P'i,\mathfrak P''_i,\mathfrak P'_j,\mathfrak P''_j). Since the order of the product
(\mathfrak N(\mathfrak A)\cdot\mathfrak N_1) is equal to the product of the orders of the subgroups (\mathfrak N(\mathfrak A){\mathfrak G}) and (\mathfrak N_1), equal respectively to (g/s) and (n), divided by the order (n/s) of the intersection
(\mathfrak N(\mathfrak A)\cap \mathfrak N_1=\mathfrak N(\mathfrak A){\mathfrak N_1}), i.e. is equal to (g), we have the equality
(\mathfrak N(\mathfrak A)\cdot \mathfrak N_1=\mathfrak G). Hence we obtain
(\mathfrak G={\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k}).

Since (\mathfrak P'_i) is an invariant (p_i)-subgroup in the group (\mathfrak G), and the order of (\mathfrak P''_i) can be divisible by only one prime number (p_i) from the set (\Pi), the order of the subgroup
(\mathfrak P_i=\mathfrak P'_i\cdot\mathfrak P''_i) can be divisible by only one prime number (p_i) from the set (\Pi), and therefore
(\Pi\cap\Pi(\mathfrak P_i)={p_i}). Hence also follows the closedness of the subgroups (\mathfrak P_i) with respect to
(\Pi\cap\Pi(\mathfrak P_i)). Thus the group (\mathfrak G) has a (\Pi)-separable basis
(\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k), which contradicts the assumption concerning this group.

If now ([\mathfrak G:\mathfrak N(\mathfrak A)_{\mathfrak G}]=1), then the subgroup (\mathfrak A) is a normal divisor of the group (\mathfrak G). The order of the factor group (\mathfrak G/\mathfrak A) is less than (g), and therefore, in view of the induction hypothesis, the factor group (\mathfrak G/\mathfrak A) has a (\Pi)-separable basis
(\mathfrak P_1/\mathfrak A,\mathfrak P_2/\mathfrak A,\ldots,\mathfrak P_k/\mathfrak A).

Since, obviously,
(\mathfrak P_i\cdot \mathfrak P_j=\mathfrak P_j\cdot \mathfrak P_i),
(\mathfrak G=\mathfrak P_1\cdot\mathfrak P_2\cdots\mathfrak P_k), and the subgroup (\mathfrak P_i),
(i=1,2,\ldots,k), is closed with respect to
(\Pi\cap\Pi(\mathfrak P_i)), while
(\Pi\cap\Pi(\mathfrak P_i)={p_i}), the collection of subgroups
(\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k) forms a (\Pi)-separable basis of the group (\mathfrak G), which contradicts the assumption about this group.

1. The order of the factor (\mathfrak N_1/\mathfrak A) is divisible by some prime number of the set (\Pi), for example by (p_1). Since the factor (\mathfrak N_1/\mathfrak A) has no proper characteristic subgroups and therefore decomposes into a direct product of mutually isomorphic simple groups, it follows from the (\Pi)-separability of the group (\mathfrak G) that the orders of the groups
   (\mathfrak P'_2\times \mathfrak P'_3\times\cdots\times \mathfrak P'_k)
   and
   (\mathfrak N_1/\mathfrak P'_2\times \mathfrak P'_3\times\cdots\times \mathfrak P'_k)
   are relatively prime.

By Schur’s theorem it follows from this that in (\mathfrak N_1) there exists a subgroup (\mathfrak A) of order equal to the order of the group
(\mathfrak N_1/\mathfrak P'2\times \mathfrak P'_3\times\cdots\times \mathfrak P'_k), and all subgroups of the group (\mathfrak N_1) of such order are conjugate to one another. If
([\mathfrak G:\mathfrak N(\mathfrak A)]\ne 1), then the order of
(\mathfrak N(\mathfrak A){\mathfrak G}) is less than (g), and consequently, by the induction hypothesis, the normalizer
(\mathfrak N(\mathfrak A)) has a (\Pi)-separable basis
(\mathfrak P''_1,\mathfrak P''_2,\ldots,\mathfrak P''_k).

As in case 1, from the assumption
([\mathfrak G:\mathfrak N(\mathfrak A)_{\mathfrak G}]\ne 1) it follows that the collection of subgroups
(\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k), where
(\mathfrak P_i=\mathfrak P'_i\cdot\mathfrak P''_i), (i=1,2,\ldots,k), forms a (\Pi)-separable basis of the group (\mathfrak G), which contradicts the assumption concerning this group.

Suppose now that (\mathfrak A) is a normal divisor of the group (\mathfrak G). In this case (\mathfrak A) cannot be a (p_1)-subgroup, since this would contradict the maximality of the special normal divisor (\mathfrak N). Moreover, for the same reason, the Sylow (p_1)-subgroup (\mathfrak B) of the normal divisor (\mathfrak A) cannot be invariant. Consequently, the order of the normalizer
(\mathfrak N(\mathfrak B){\mathfrak G}) is less than (g), and therefore, in view of the induction hypothesis, the subgroup
(\mathfrak N(\mathfrak B)) has a (\Pi)-separable basis
(\mathfrak P''_1,\mathfrak P_2,\ldots,\mathfrak P_k).

Putting (\mathfrak P_1=\mathfrak P_1''\cdot\mathfrak A), it is easy to show that the collection of subgroups (\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k) forms a (\Pi)-separable basis of the group (\mathfrak G).

Indeed, the permutability of the subgroups (\mathfrak P_i) and (\mathfrak P_j) is obvious, and the relation (\mathfrak G=\mathfrak P_1\cdot\mathfrak P_2\cdots\mathfrak P_k) follows from the equality (\mathfrak G=\mathfrak A\cdot\mathfrak N(\mathfrak B){\mathfrak G}). By virtue of the equality of indices
[
[\mathfrak G:\mathfrak N(\mathfrak B)]=[\mathfrak A:\mathfrak N(\mathfrak B){\mathfrak A}]
]
the order of (\mathfrak N(\mathfrak B)) is divisible by the number
[
p_1^{\alpha_1}\cdot p_2^{\alpha_2}\cdots p_k^{\alpha_k},
]
where (\alpha_i) is the highest power to which the prime number (p_i) occurs in (g). Hence it follows that the subgroups (\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k) are closed in the group (\mathfrak G) with respect to the corresponding sets (\Pi\cap\Pi(\mathfrak P_1''), \Pi\cap\Pi(\mathfrak P_2),\ldots,\Pi\cap\Pi(\mathfrak P_k)), and consequently (since (\Pi\cap\Pi(\mathfrak A)={p_1})), that the subgroups (\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k) are closed with respect to the sets (\Pi\cap\Pi(\mathfrak P_i)), (i=1,2,\ldots,k). A contradiction to the assumption concerning the group (\mathfrak G) has again been obtained. Theorem 1 is proved.

Theorem 2. A locally (\Pi)-separable group (\mathfrak G), i.e. a locally finite group every finite subgroup of which is (\Pi)-separable, possessing a normal series with finite factors and satisfying the minimal condition for subgroups, has at least one (\Pi)-separable basis.

Proof. Since the group (\mathfrak G) is a finite extension of a maximal complete normal divisor (\mathfrak A) (see 2, Theorem 2), it can be represented in the form of a product (\mathfrak G=\mathfrak A\cdot\mathfrak B), where (\mathfrak B) is some finite subgroup. By virtue of Theorem 1 the finite group (\mathfrak B) has a (\Pi)-separable basis (\mathfrak P_1',\mathfrak P_2',\ldots,\mathfrak P_k'), and if the prime number (p_i\in\Pi\cap\Pi(\mathfrak G)={p_1,p_2,\ldots,p_k}) does not divide the order of the group (\mathfrak B), then the subgroup (\mathfrak P_i') coincides with the identity.

Next denote by (\mathfrak P_1'', \mathfrak P_2'',\ldots,\mathfrak P_{k-1}''), respectively, the Sylow (p_i)-subgroups, (i=1,2,\ldots,k-1), of the group (\mathfrak A), and by (\mathfrak P_k'') the Sylow (\Pi')-subgroup of this group, where (\Pi') is the set of prime divisors of the orders of elements of the group (\mathfrak A) distinct from the numbers (p_1,p_2,\ldots,p_{k-1}). From the commutativity of the group (\mathfrak A) it follows that the collection of subgroups (\mathfrak P_1'',\mathfrak P_2'',\ldots,\mathfrak P_k'') forms its (\Pi)-separable basis. But then, putting (\mathfrak P_i=\mathfrak P_i'\cdot\mathfrak P_i''), we obtain, as is easy to show, a (\Pi)-separable basis of the group (\mathfrak G): (\mathfrak P_1,\mathfrak P_2,\ldots,\mathfrak P_k).

Theorem 2 is proved.

Let us note that Theorem 3 of the work ((^3)) is a special case of Theorem 2.

In conclusion I take this opportunity to express my gratitude to Prof. N. S. Chernikov for his suggestions.

Molotov State University
named after A. M. Gorky

Received
4 VI 1956

CITED LITERATURE

1. S. A. Chunikhin, DAN, 95, No. 4 (1954).
2. B. I. Plotkin, DAN, 107, No. 5, 648 (1956).
3. P. A. Gol’berg, Matem. sborn., 32 (74), 465 (1953).