Maximal subgroups and the Jordan-Hölder Theorem

Lafuente, Julio

doi:10.1017/s1446788700030846

Cited by 5 publications

(4 citation statements)

References 13 publications

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“…Nevertheless, the arguments we have presented here to prove Theorem A(a) are exactly those to prove that M is an M-set. It must be observed that in [13] it is proved that the bijection mentioned in Theorem A(a) is unique.…”

Section: Is X-frattini If and Only If M^/m^^ Is X-frattini; (Iii) / /mentioning

confidence: 99%

“…Given a modular lattice if, J. Lafuente in [13] introduced the concept of M-set in 5£ and he proved a general Jordan-Holder theorem in modular lattices with an M-set.…”

Section: Is X-frattini If and Only If M^/m^^ Is X-frattini; (Iii) / /mentioning

confidence: 99%

“…Conversely, suppose that X is a set of monolithic maximal subgroups of G such that the set of X-supplemented chief factors M is an M-set in M. If On page 361 of [13], Lafuente wonders about the precise condition on any set of maximal subgroups X of G such that the set of chief factors of G complemented by at least one element U e X is an M-set in Jf. What we have shown above is that ///-solid sets of maximal subgroups are the answer to Lafuente's question for sets of monolithic maximal subgroups if complementation is changed to supplementation.…”

Section: Is X-frattini If and Only If M^/m^^ Is X-frattini; (Iii) / /mentioning

confidence: 99%

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The Jordan-Hölder theorem and prefrattini subgroups of finite groups

Ballester‐Bolinches

Ezquerro

1995

Glasgow Math. J.

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Introduction. All groups considered are finite. In recent years a number of generalizations of the classic Jordan-Holder Theorem have been obtained (see [7], Theorem A.9.13): in a finite group G a one-to-one correspondence as in the JordanHolder Theorem can be defined preserving not only G-isomorphic chief factors but even their property of being Frattini or non-Frattini chief factors. In [2] and [13] a new direction of generalization is presented: the above correspondence can be defined in such a way that the corresponding non-Frattini chief factors have the same complement (supplement).In this paper we present a necessary and sufficient condition, named (JH), for a set X of monolithic maximal subgroups of a group G to verify this wider version of the Jordan-Holder Theorem: the Jordan-Holder bijection can be defined in such a way that the corresponding chief factors are G-isomorphic, have a supplement in X at the same time and, in this case, it can be chosen to be a common supplement in X. It is remarkable that (JH) is not only a sufficient condition but is indeed necessary.On the other hand, W. Gaschiitz introduced in [10] a conjugacy class of subgroups of a finite soluble group called prefrattini subgroups. They form a characteristic conjugacy class of subgroups which cover the Frattini chief factors and avoid the complemented ones. These results were generalized by Hawkes [11] and Fbrster [8] in the soluble case. In [1], the authors introduced the concept of a system of maximal subgroups. These systems can be used to select maximal subgroups in order to define prefrattini subgroups similar to those of Gaschiitz, Hawkes and Forster in the general non-soluble case. They enjoy most of the properties of the soluble case except the Cover and Avoidance Property and conjugacy. In fact, conjugacy characterizes solubility.Another generalization of Gaschiitz's work in the soluble universe is due to Kurzweil [12]. He introduced the //-prefrattini subgroups of a soluble group G, where H is a subgroup of G. The //-prefrattini subgroups are conjugate in G and they have the Cover and Avoidance Property; if H = 1 they coincide with the classical prefrattini subgroups of Gaschiitz and if § is a saturated formation and H is an g-normalizer of G the //-prefrattini subgroups are those described by Hawkes.Tomkinson in [14] extended the results of Gaschiitz and Hawkes to a class It of locally finite groups with a satisfactory Sylow structure. The intersection of II with the class of all finite groups is just the class of all finite soluble groups.Our aim here is to present all the results of the finite universe in a more unified setting. Our approach is based on Tomkinson's ideas. Using a property, denoted by (*), which is slightly stronger than (JH), we can define X-prefrattini subgroups of a finite

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Section: Is X-frattini If and Only If M^/m^^ Is X-frattini; (Iii) / /mentioning

confidence: 99%

“…Given a modular lattice if, J. Lafuente in [13] introduced the concept of M-set in 5£ and he proved a general Jordan-Holder theorem in modular lattices with an M-set.…”

Section: Is X-frattini If and Only If M^/m^^ Is X-frattini; (Iii) / /mentioning

confidence: 99%

Section: Is X-frattini If and Only If M^/m^^ Is X-frattini; (Iii) / /mentioning

confidence: 99%

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The Jordan-Hölder theorem and prefrattini subgroups of finite groups

Ballester‐Bolinches

Ezquerro

1995

Glasgow Math. J.

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“…Our primary objective is to generalise further the version of the Jordan-Hölder Theorem for chief series of L established in [4]. Our result could probably be obtained from [2], but we prefer to follow the approach adopted for groups (though in a more general context) in [1] as interesting results and concepts are obtained on the way.…”

Section: Introductionmentioning

confidence: 99%