Finite biprimary groups with all 3-maximal subgroups $${\mathfrak{U}}$$ U -subnormal

Kovaleva, V. A.; Yi, Xiaolan

doi:10.1007/s10474-015-0498-5

Cited by 7 publications

(2 citation statements)

References 9 publications

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“…The relationship between n-maximal subgroups (where n > 1) of a group G and the structure of G was studied by many authors (see, in particular, the recent papers [13,14,15,16,17,18,19,20] and Chapter 4 in the book [11]). One of the first results in this direction were obtained by Huppert [21].…”

Section: Introductionmentioning

confidence: 99%

Finite groups whose n-maximal subgroups are σ-subnormal

Guo

Skiba

2018

Sci. China Math.

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Let σ = {σ i |i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hall σ-set of G if every member = 1 of H is a Hall σ i -subgroup of G, for some i ∈ I, and H contains exact one Hall σ i -subgroup of G for every σ i ∈ σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set set H such that HAwhere M i is a maximal subgroup of M i−1 , i = 1, 2, . . . , n, then M is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal (σ-quasinormal, respectively) in G but, in the case n > 1, some (n − 1)maximal subgroup is not σ-subnormal (not σ-quasinormal, respectively)) in G, we write m σ (G) = n (m σq (G) = n, respectively).In this paper, we show that the parameters m σ (G) and m σq (G) make possible to bound the σ-nilpotent length l σ (G) (see below the definitions of the terms employed), the rank r(G) and the number |π(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when m σ (G) = |π(G)|. Some known results are generalized.

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Section: Introductionmentioning

confidence: 99%

Finite groups whose n-maximal subgroups are σ-subnormal

Guo

Skiba

2018

Sci. China Math.

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“…The relations between n-maximal subgroups (for n > 1) of G and the structure of G was studied by many authors (see, for example, [7][8][9][10][11][12] and Chapter 4 in the book [4]). One of the earliest results in this direction were obtained by Huppert [13], who proved that if every 2-maximal subgroup of G is normal, then G is supersoluble; if every 3-maximal subgroup of G is normal in G, then G is a soluble group of rank(G) at most two.…”

mentioning

confidence: 99%

On an open problem of Guo-Skiba

Guo

2016

Front. Math. China

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Let σ = {σ i |i ∈ I} be some partition of the set P of all primes, that is, P = i∈I σ i and σ i ∩ σ j = ∅ for all i = j. Let G be a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ i -subgroup of G and H contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ(G). G is said to be a σ-group if it possesses a complete Hall σ-set. A σ-group G is said to be σ-dispersive provided G has a normal series 1 = G 1 < G 2 < · · · < G t < G t+1 = G and a complete Hall σ-set {H 1 , H 2 , · · · , H t } such that G i H i = G i+1 for all i = 1, 2, . . . t. In this paper, we give a characterizations of σ-dispersive group, which give a positive answer to an open problem of Skiba in the paper [1].

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Finite Groups with Given Systems of K-U-Subnormal Subgroups

Kovaleva

2016

Ukr Math J

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Finite biprimary groups with all 3-maximal subgroups $${\mathfrak{U}}$$ U -subnormal

Cited by 7 publications

References 9 publications

Finite groups whose n-maximal subgroups are σ-subnormal

Finite groups whose n-maximal subgroups are σ-subnormal

On an open problem of Guo-Skiba

Finite Groups with Given Systems of K-U-Subnormal Subgroups

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