Maximal nonnormal chains in finite groups

Spencer, Armond E.

doi:10.2140/pjm.1968.27.167

Cited by 15 publications

(9 citation statements)

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“…Some of the same structure was noted by the author [6,Theorem 4] under the weaker hypothesis, h(G) = n. To see that this is truly a weaker hypothesis, consider the group G given by G = S 3 IZ X . \GI = 72, and the Sylow-3-subgroup is a minimal normal subgroup.…”

Section: Any Case H(g/a)mentioning

confidence: 67%

“…The proof is by induction on n. For n = 2, the theorem follows from [6,Theorem 5], so assume the theorem is true for all groups K satisfying 2 ^ h(K) ^ (n -1), and is false for some group K satisfying h(K) = n. Among such groups, let G be one of minimal order. We show that such a group G does not exist.…”

Section: / / G Is a Finite Solvable Group Such That 2 ^ H(g) G N Thementioning

confidence: 99%

“…This follows from (2) and (4), and the fact that the next to last entry in an /i-chain for G is cyclic, primary, and not subnormal. Here /j-chain refers to any upper chain in G of length h(G) with only its terminal entry subnormal in G. (6) For each r, h(GjG {r) ) = r or 1. This is certainly true for r = 1 and r = n, so suppose /2(G/G (r+1) ) = r + 1, and consider GlG (r) .…”

Section: Any Case H(g/a)mentioning

confidence: 99%

“…Let JV be the product of all normal Sylow subgroups of G and let H be a complement to N. By [6,Theorem 4], N has the required structure, and if n(H) 2: 2, then | H | is square free. All that remains to be shown is that in the case n(H) ^ 2, H is cyclic.…”

Section: Ifg Is a Finite Solvable Group Such That H(g) = N(g) ^ 2 Thmentioning

confidence: 99%

“…We see this by looking at a chain thru N(Q) and PQ to Q. From [6,Theorem 4] it follows that if n(H) ^ 2, then | H | is square free, so it remains to show that H is abelian. We consider two cases: Case 1.…”

Section: Ifg Is a Finite Solvable Group Such That H(g) = N(g) ^ 2 Thmentioning

confidence: 99%

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Finite groups with short nonnormal chains

Spencer

1974

J. Aust. Math. Soc.

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This note is a continuation of the author's work [6], describing the structure of a finite group given some information about the distribution of the subnormal subgroups in the lattice of all subgoups.DEFINITION. An upper chain of length n in the finite group G is a sequence of subgroups of G; G = Go > G1 > … > Gn, such that for each i, Gi is a maximal subgroup of Gi-1. Let h(G) = n if every upper chain in G of length n contains a proper ( ≠ G) subnormal entry, and there is at least one upper chain in G of length (n – 1) which contains no proper subnormal entry.

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Section: Any Case H(g/a)mentioning

confidence: 67%

Section: / / G Is a Finite Solvable Group Such That 2 ^ H(g) G N Thementioning

confidence: 99%

Section: Any Case H(g/a)mentioning

confidence: 99%

Section: Ifg Is a Finite Solvable Group Such That H(g) = N(g) ^ 2 Thmentioning

confidence: 99%

Section: Ifg Is a Finite Solvable Group Such That H(g) = N(g) ^ 2 Thmentioning

confidence: 99%

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Finite groups with short nonnormal chains

Spencer

1974

J. Aust. Math. Soc.

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Finite groups

Chunikhin¹,

Shemetkov²

1973

J Math Sci

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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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Maximal nonnormal chains in finite groups

Cited by 15 publications

References 4 publications

Finite groups with short nonnormal chains

Finite groups with short nonnormal chains

Finite groups

Finite groups whose n-maximal subgroups are σ-subnormal

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