Finite Groups with Subnormal Schmidt Subgroups

Knyagina, V. N.; Монахов, В. С.

doi:10.1023/b:simj.0000048922.59466.20

Cited by 23 publications

(15 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Since P (respectively Q) is maximal -subgroup (respectively -subgroup), P (respectively Q) is normal in G. Proof. If G is finite, the result follows from the main theorem of [6]. Suppose now that G is infinite.…”

Section: Let G Be a Group Whose Non-abelian Subgroups Are Subnormal mentioning

confidence: 94%

See 1 more Smart Citation

On the structure of groups whose non-abelian subgroups are subnormal

Kurdachenko¹,

Atlıhan

Semko³

2014

Open Mathematics

View full text Add to dashboard Cite

show abstract

Section: Let G Be a Group Whose Non-abelian Subgroups Are Subnormal mentioning

confidence: 94%

“…Knyagina and V.S. Monakhov [6] began studying finite groups with subnormal Schmidt subgroups. Later, V.A.…”

Section: Introductionmentioning

confidence: 99%

On the structure of groups whose non-abelian subgroups are subnormal

Kurdachenko¹,

Atlıhan

Semko³

2014

Open Mathematics

View full text Add to dashboard Cite

show abstract

“…Schmidt groups have found numerous applications to problems in group theory (see, e.g., [5][6][7][8][9][10][11][12]). To our knowledge, possibilities for using Schmidt groups in studies of finite groups were first mentioned in [5,Chap.…”

Section: Introductionmentioning

confidence: 99%

“…In [9,11] is a review of results on Schmidt groups and their applications in group theory. Thus, [8,12] treat of properties of a finite non-nilpotent group all Schmidt subgroups of which are subnormal. In [8], it was proved that such a group is metanilpotent, and in [12], it was stated that the derived subgroup of the group is nilpotent.…”

Section: Introductionmentioning

confidence: 99%

Finite groups with subnormal Schmidt subgroups

Vedernikov

2007

Algebra Logic

View full text Add to dashboard Cite

“…LEMMA 1 [3]. If K and D are subgroups of G, the subgroup D is normal in K, and K/D is an S p,q -subgroup, then the minimal complement L to D in K possesses the following properties:…”

mentioning

confidence: 99%

Finite groups with seminormal Schmidt subgroups

Knyagina

Монахов

2007

Algebr Logic

Self Cite

View full text Add to dashboard Cite

A non-nilpotent finite group whose proper subgroups are all nilpotent is called a Schmidt group. A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB 1 is a proper subgroup of G, for every proper subgroup B 1 of B. Groups that contain seminormal Schmidt subgroups of even order are considered. In particular, we prove that a finite group is solvable if all Schmidt {2, 3}-subgroups and all 5-closed {2, 5}-Schmidt subgroups of the group are seminormal; the classification of finite groups is not used in so doing. Examples of groups are furnished which show that no one of the requirements imposed on the groups is unnecessary.We deal with finite groups only. A non-nilpotent group whose proper subgroups are all nilpotent is called a Schmidt group. The Schmidt groups were brought in sight in [1] where it was proved that such are biprimary, and also that in each of these groups, one Sylow subgroup is normal and the other is cyclic. A detailed review of the structure of Schmidt groups and their applications in finite group theory can be found in [2].A subgroup A is said to be seminormal in a group G if there exists a subgroup B such that G = AB and AB 1 is a proper subgroup of G, for every proper subgroup B 1 of B. In this setting, the subgroup B is called a supercomplement to the subgroup A. Obviously, a subgroup of prime index is always seminormal. A quasinormal subgroup (i.e., one that commutes with all subgroups of the group) will be seminormal. In a simple group P SL(2, 5), the subgroup A isomorphic to an alternating group of degree 4 is a seminormal Schmidt subgroup, but A is neither quasinormal nor subnormal.In the present paper, we look at groups that contain seminormal Schmidt subgroups of even order. We will argue for the following two theorems. THEOREM 1. If A is a seminormal Schmidt subgroup of a group G, and the subgroupTHEOREM 2. If all Schmidt {2, 3}-subgroups of a group G are seminormal, then the group G is 3-solvable.COROLLARY. If all Schmidt {2, 3}-subgroups and all 5-closed Schmidt {2, 5}-subgroups of a group G are seminormal then the group G is solvable.In proving the theorems and the corollary, we will not appeal to the classification of finite simple groups. * Supported by BelFBR grant Nos. F05-341 and F06MS-017.1 Gomel Engineering Institute under the Belarussian Ministry of Extraordinary Situations; knyagina@inbox.ru.

show abstract

Finite Groups with Subnormal Schmidt Subgroups

Cited by 23 publications

References 3 publications

On the structure of groups whose non-abelian subgroups are subnormal

On the structure of groups whose non-abelian subgroups are subnormal

Finite groups with subnormal Schmidt subgroups

Finite groups with seminormal Schmidt subgroups

Contact Info

Product

Resources

About