The normal index of maximal subgroups in finite groups

Beidleman, James C.; Spencer, Armond E.

doi:10.1215/ijm/1256052386

Cited by 13 publications

(12 citation statements)

References 2 publications

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“…Next we establish how the ideal index behaves with respect to factor algebras. The following result, or rather its corollary, is analogous to [5,Lemma 2], though our proof is somewhat different. Proof.…”

Section: Ideal Completions and The Ideal Indexmentioning

confidence: 77%

The index complex of a maximal subalgebra of a Lie algebra

Towers

2011

Proceedings of the Edinburgh Mathematical Society

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Let M be a maximal subalgebra of the Lie algebra L. A subalgebra C of L is said to be a completion for M if C is not contained in M but every proper subalgebra of C that is an ideal of L is contained in M . The set I(M ) of all completions of M is called the index complex of M in L. We use this concept to investigate the influence of the maximal subalgebras on the structure of a Lie algebra, in particular finding new characterisations of solvable and supersolvable Lie algebras.

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Section: Ideal Completions and The Ideal Indexmentioning

confidence: 77%

The index complex of a maximal subalgebra of a Lie algebra

Towers

2011

Proceedings of the Edinburgh Mathematical Society

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“…(6) Letting G('P|) be the smallest normal subgroup of G for which the factor group is an abelian /»¡-group, we note first that G[P0 = G for each i^r. If not, it would be nilpotent by hypothesis, hence /»¡-nilpotent, and its characteristic /^-complement would then be a normal /»¡-complement ofG, in contradiction to (2). It now follows from one of the basic transfer theorems (see [16, 13.5.2]) that 1 =G/G('Pl)~P¡/7J¡ n G' so that Pi^G' for each i^r.…”

Section: Corollarymentioning

confidence: 95%

“…For some i+r, P¡ is not minimal normal in G. Let M =£ 1 be a normal subgroup of G properly contained in P¡. Then G/M is not /?¡-nilpotent; for if it were, say with normal /»¡-complement T/M, then T, being a proper normal subgroup of G, would be nilpotent, hence /»¡-nilpotent, and its characteristic /»¡-complement would then be a normal /»¡-complement of G in contradiction to (2). Also, the other hypotheses hold for G/A7 so that by induction, for each i+r, pr divides d^ = Ylbji=1(p'i-\), where |G/A7| = n¡ = i pV-Hence, pr also divides dp=Y\%i (p{-\) for each i<£r.…”

Section: Corollarymentioning

confidence: 98%

“…Let g be a /?r-Sylow subgroup of G. From (4), |ö/corG Q\ =pn so since ar> 1, corG Qj=\. Now, G/corG Q is non-/?rnilpotent for i^r; otherwise, if F/corG Q were a normal/»¡-complement, Fwould be a normal/»¡-complement of G in contradiction to (2). Again, the other hypotheses hold for G/corG Q so that, by induction, since all the Pi divide |G/corG Q\, we have the result as in Case 1.…”

Section: Corollarymentioning

confidence: 98%

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The influence on a finite group of the cofactors and subcofactors of its subgroups

Nyhoff¹

1971

Trans. Amer. Math. Soc.

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Abstract. The effect on a finite group G of imposing a condition 6 on its proper subgroups has been studied by Schmidt, Iwasawa, Itô, Huppert, and others. In this paper, the effect on G of imposing 9 on only the cofactor H/cora H (or more generally, the subcofactor ///scorG H) of certain subgroups H of G is investigated, where cor0 H (scor0 H) is the largest G-normal (G-subnormal) subgroup of H. It is shown, for example, that if (a) ///scorc H is /»-nilpotent for all self-normalizing H -Sylows of G are abelian, then in either case, G has a normal p-subgroup P for which G/P is p-nilpotent. Results of this type are also derived for 6 = nilpotent, nilpotent of class an, solvable of derived length g«, o-Sylow-towered, supersolvable. In some cases, additional structure in G is obtained by imposing 8 not only on these "worst" parts of the "bad" subgroups of G (from the viewpoint of normality), but also on the "good" subgroups, those which are normal in G or are close to being normal in that their cofactors are small.Finally, this approach is in a sense dualized by an investigation of the influence on G of the outer cofactors of its subgroups. The consideration of nonnormal outer cofactors is reduced to that of the usual cofactors. The study of normal outer cofactors includes the notion of normal index of maximal subgroups, and it is proved, for example, that G is p-solvable iff the normal index of each abnormal maximal subgroup of G is a power of p or is prime to p.There are a number of theorems which describe the effect on a finite group G of a condition 6 imposed on its proper subgroups. For example, Schmidt [15] and Iwasawa [12] have shown that if every proper subgroup of a finite group G is nilpotent, then G is solvable, and among other things (see Theorem 2-A), if G itself is nonnilpotent, then \G\ =paq" for distinct primes p and q, G has a normal /j-Sylow subgroup and cyclic í7-Sylow subgroups. Huppert [10] and Doerk [5] have obtained corresponding results for the case where the proper subgroups of G are

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“…Deskins [2] showed that is solvable if and only if ( : ) = | : | for every maximal subgroup of . The investigations on the normal index have been developed by many scholars; see [3][4][5][6][7]. But the earlier results concern the cases where is either the largest prime dividing | | or an odd prime.…”

Section: Introductionmentioning

confidence: 99%