Nonabelian crowns and Schunck classes of finite groups

Lafuente, Julio

doi:10.1007/bf01198125

Cited by 22 publications

(10 citation statements)

References 8 publications

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“…The notion of crown associated with a non-Frattini chief factor was introduced for finite solvable groups (so for abelian chief factors) by Gaschütz in [8] and extended to arbitrary finite groups (so to non-abelian chief factors) in [13]. In [6,Theorem 11] the authors show that if G is a finitely generated profinite group, then for any non-Frattini chief factor A, the maximum size of a crown-based power associated with L G (A), which is an epimorphic image of G, is δ G (A).…”

Section: Apfg Groupsmentioning

confidence: 99%

The generation of the augmentation ideal in profinite groups

Damian

2011

Isr. J. Math.

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We find a formula for computing the minimum number of generators for the augmentation ideal in the profinite setting; this is a generalisation of a result obtained in the finite case by Cossey, Gruenberg and Kovács.We then consider probabilistic questions connected with the generation of the augmentation ideal and we define the class of APFG groups as those profinite groups for which the probability of generating the augmentation ideal with t random elements is non-zero for some t ∈ N. We give a characterisation of these groups which allows us easily to compare APFG groups with PFG groups and we show that PFG groups are APFG but we give examples showing that this is a strict inclusion.

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Section: Apfg Groupsmentioning

confidence: 99%

The generation of the augmentation ideal in profinite groups

Damian

2011

Isr. J. Math.

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“…consists of maximal subgroups U of G complementing a chief factor of G between R and C; in view of the structure of crowns (cf. 2.4 in [7]), this means that U complements a minimal normal subgroup of G/COKG (U) andR<Con G (U) < C. Next, let ^c/R t> e t n e s e t °f all maximal subgroups complementing a chief factor between R and C. Recall that all chief factors X/Y of G between R and C are isomorphic (although all of them are (7-isomorphic only if C/R is abelian or is itself a chief factor of G; however, they are always similar in the sense of 53.11 of [15], and G-connected/G-related in the sense of [13] and [7]. Observe that all these chief factors X/ Y are complemented in G by a maximal subgroup, except those for which X = C and G/Y <$.…”

Section: Xr/yrmentioning

confidence: 99%

“…for Jty to satisfy (M2). Before stating (#), we recall from Baer and Forster [2], Forster [7], Lafuente [13], the definition of the crown C/R of a group G associated with its non-Frattini chief factor X/ Y:…”

Section: We Set •Andkh = {Xe And\k < X < H} and And> Kji = {X/ye And\k < Y Amentioning

confidence: 99%

“…(However, for such a correspondence, corresponding chief factors are not normally Gisomorphic, but only G-connected as defined by the author in [ 13] and, independently, by Forster in [7] ((7-related) and [2] (G-verwandt). )…”

mentioning

confidence: 99%

“…Basic properties of crowns are described in [2,7,13], and will be used without further reference. From such properties the following is immediate.…”

Section: We Set •Andkh = {Xe And\k < X < H} and And> Kji = {X/ye And\k < Y Amentioning

confidence: 99%

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Maximal subgroups and the Jordan-Hölder Theorem

Lafuente

1989

J Aust Math Soc A

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In this note we present a general Jordan-Holder type theorem for modular lattices and apply it to obtain various (old and new) versions of the Jordan-Holder Theorem for finite groups. Isbell [ 10] has observed that the Jordan-Holder Theorem may be derived from the Zassenhaus Theorem, and that this yields a uniqueness statement for the correspondence given by the Jordan-Holder Theorem. This result, however, does not give the various versions of the Jordan-Holder Theorem for finite groups that have received some interest more recently, for example, the one that states that for any two chief series of a finite group a correspondence can be found associating Frattini chief factors with Frattini chief factors and non-Frattini ones with non-Frattini ones. Such a theorem was first published by Carter, Fischer and Hawkes [4] for finite soluble groups, and for finite groups in general in the author's [12], with a different approach by Forster in [7] (see also Chapter 1 of [2]). Further, Barnes proved that in soluble groups corresponding complemented (which, for finite soluble groups, means nonFrattini) chief factors have a common (maximal) complement. On the other hand, for arbitrary finite groups the number of complemented chief factors in a given chief series can depend on the series (see Baer and Forster [2] or Kovacs and Newman f 11] for examples).

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Classes of generalized nilpotent and soluble groups

Asiáin¹

1996

Sib Math J

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Nonabelian crowns and Schunck classes of finite groups

Cited by 22 publications

References 8 publications

The generation of the augmentation ideal in profinite groups

The generation of the augmentation ideal in profinite groups

Maximal subgroups and the Jordan-Hölder Theorem

Classes of generalized nilpotent and soluble groups

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