A Criterion for the $\sigma$ -Subnormality of a Subgroup in a Finite $3^{'}$ -Group
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Criterion for the σ-Subnormality of a Subgroup in a Finite 3 -Group S. F. Kamornikov1* and
V. N. Tyutyanov2**
1
Francisk Skorina Gomel State University, 104 Sovetskaya str., Gomel, 246019 Republic of Belarus 2 Gomel Branch of the International University «MITSO», 46 a October Ave., Gomel, 246029 Republic of Belarus Received September 23, 2019; revised September 23, 2019; accepted June 29, 2020
Abstract—For any partition σ of the set P of all primes, it is proved that if a subgroup H of a finite 3 -group G is σ-subnormal in < H, H x > for any x ∈ G, then H is σ-subnormal in G. DOI: 10.3103/S1066369X20080046 Key words: finite group, σ-subnormal subgroup, subnormal subgroup, Suzuki group.
INTRODUCTION The following Wielandt criterion of subnormality is well known [1]: a subgroup H of a finite group G is subnormal if and only if it is subnormal in < H, H x > for any x ∈ G. This result initiated the respective problem for σ-subnormal subgroups of a finite group asked by A.N. Skiba in [2] under number 19.84 (see also problem 7.5 from [3]). Let for any x ∈ G a subgroup H of a group G be σ-subnormal in < H, H x >. Is it true then that subgroup H is σ-subnormal in G? In the case when G is a finite 3 -group, positive answer is given in the theorem in the present paper. The concept of a σ-subnormal subgroup was suggested by A.N. Skiba in [5], it develops the idea of a subnormal subgroup from [4]. This concept is based on the following definitions. Let P be the set of all primes, π ⊆ P and π = P \ π. For a natural number n, by π(n) we denote the set of all primes which divide n; in particular, π(G) = π(|G|) is the set of all primes dividing the order |G| of a group G. Everywhere further, σ is some partition of the set P by pairwise disjoint subsets σi (i ∈ I), i. e. P = i∈I σi and σi ∩ σj = ∅ for all i = j. Following [5], we will say that a group G is σ-primary, if G is a σi -group for some i ∈ I. A subgroup H of group G is called σ-subnormal, if there exists a chain of subgroups H = H0 ⊆ H1 ⊆ . . . ⊆ Hn = G such that for any i = 1, 2, . . . , n either subgroup Hi−1 is normal in Hi or group Hi /CoreHi (Hi−1 ) is σ-primary. It is clear that subgroup H is subnormal in G if and only if it is σ-subnormal in G for the minimal decomposition σ = {{2}, {3}, {5}, . . .}. Our main objective is the proof of the following theorem.
Theorem. Let σ be some partition of the set P of all primes. A subgroup H of a finite 3 -group G is σ-subnormal in G if and only if H is σ-subnormal in < H, H x > for any x ∈ G. The keys to the proof of this theorem are the proposition describing the construction of minimal counterexample to the problem 19.84 together with the paper by M. Suzuki on simple nonabelian 3 -groups. * **
E-mail: [email protected] E-mail: [email protected]
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A CRITERION FOR THE σ-SUBNORMALITY
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1. DEFINITIONS AND PRELIMINARY RESULTS In this article we consider only finite groups and use definitions and notation from [7]. As for the terminology of the theory of σ-subnormal subgroups, the reader may refer [5]. A
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