1,720,983 research outputs found
Gruppi con pochi sottogruppi non quasinormali
Full text linkLet be a group. A subgroup of is said to be quasinormal if for every subgroup of , and is called quasi-Hamiltonian if all its subgroups are quasinormal. The structure of quasi-Hamiltonian groups has been described by {\it K. Iwasawa} [in J. Fac. Sci., Univ. Tokyo, Sect. I 4, 171-199 (1941; Zbl 0061.025) and Jap. J. Math. 18, 709-728 (1943; Zbl 0061.025)]. If is a group, let denote the subgroup generated by all subgroups of which are not quasinormal. Then is quasi-Hamiltonian if and only if , and it is easy to show that is generated by all cyclic non-quasinormal subgroups of .\par The author studies the class of all groups for which is a proper subgroup. The corresponding problem for the subgroup generated by all non-normal subgroups was considered by {\it D. Cappitt} [J. Algebra 17, 310-316 (1971; Zbl 0232.20067)]. Clearly every -group is generated by cyclic quasinormal subgroups, and in particular it is locally nilpotent. The author proves that non-periodic -groups are quasi-Hamiltonian. The investigation of periodic -groups can be reduced to the case of a -group ( prime), and the description of -groups in the class is obtained. In particular, it is shown that a -group of infinite exponent is in the class if and only if the subgroup generated by all non-normal subgroups of is properly contained in . Finally, the author proves that if is an -group whose Sylow 2-subgroup is quasi-Hamiltonian, then is metabelian.
[F.de Giovanni (Napoli)
A Simple Proof of Baer's and Sato's Theorems on Lattice Isomorphisms between Groups
No full textA group is said to be an M\sp*-group if all subgroups of are quasinormal and is quaternionfree. Using Iwasawa's characterization of M\sp*-groups the author gives an elegant and unified proof of the following theorem: If is an M\sp*-group, then there exists an abelian group such that the lattices of subgroups of and are isomorphic.
[J.Chvalina (Brno)
On the Poset of Conjugacy Classes of Subgroups of Finite p-Groups
No full textFor a finite group , let denote the poset of conjugacy classes of subgroups of . The ordering is iff for some . Let be a finite group such that and are order-isomorphic. Let is a -group. It is known [{it R. Brandl}, Commun. Algebra 20, No. 10, 3043-3054 (1992; Zbl 0767.20008)] that is a -group. Moreover, if is abelian or metacyclic, then [see {it L. Verardi} and the reviewer, Glasg. J. Math. 35, No. 3, 339-344 (1993; Zbl 0846.20020)].par Let be an order-isomorphism. It is shown in this interesting paper that maps (conjugacy classes of) normal subgroups of onto normal subgroups of . This answers a question of L. Verardi and the reviewer. Indeed, a stronger result is proved concerning -groups with a -group of operators. The specialization yields, for example, the known result that a projectivity between two -groups maps some principal series of onto a principal series of . Another consequence of the main theorem is that the classes of -groups of maximal nilpotency class, and of powerful -groups are determined by their posets of subgroups. Moreover, if is a modular -group, then .par Generally speaking, the paper supports the philosophy that posets of subgroups, although very much smaller, behave much better than the lattices of all subgroups, but seem to contain enough information to recover many properties of the group under consideration.
[R.Brandl (Würzburg)
Modular sublattices in a finite p-group
No full textDer Verf. studiert den Durchschnitt aller maximalen modularen Teilverbände des Untergruppenverbandes {\goth L}(G) einer Gruppe und nennt ihn {\goth Z}(G), da er genau die von Zacher bereits 1955 studierten Untergruppen enthält, die mit jedem modularen Teilverband einen modularen Teilverband von {\goth L}(G) erzeugen. Natürlich gehören 1 und zu {\goth Z}(G), und der Verf. zeigt, dass eine der folgenden drei Aussagen gilt, wenn eine endliche -Gruppe mit und M\in{\goth Z}(G) mit ist:\par (1) {\goth L}(G) ist modular.\par (2) für jedes mit {\goth L}(H) nicht modular.\par (3) enthält jede nicht permutable Untergruppe von .\par Der Verf. gibt Beispiele für -Gruppen mit Elementen von {\goth Z}(G), die Eigenschaft (2) bzw. (3) haben. Ob aber jedes solche zu {\goth Z}(G) gehört, bleibt offen.
[R.Schmidt (Kiel)
Projektivitaeten zwischen abelschen und nichtabelschen Gruppen
No full textIn 1951 S. Sato showed that the lattice of subgroups of a modular group with elements of infinite order is isomorphic to the one of a convenient abelian group. Recently in the last part of Sato's work some inexactitudes were found which could question the validity of the result. \par In this paper a new proof of that theorem is provided. Included are also two results on modular groups with elements of infinite order. Namely that any such group can be embedded in a modular group whose torsion- subgroup is divisible and that in case the torsion-subgroup is divisible a modular group splits into the semidirect product of its torsion- subgroup by a cyclic, or locally cyclic, group
Finite groups whose poset of conjugacy classes of subgroups is isomorphic to the one of an abelian group
No full textGroups in which every subgroup is f-subnormal
No full textA subgroup of a group is called -subnormal in , if there is a finite sequence such that the predecessor is normal in the following term whenever the index is infinite. It follows from results of Lennox and Stonehewer that finitely generated groups of the title are finite-by-nilpotent. Two results for the general case as examples: is finite-by-solvable, every subgroup of is subnormal and is finite-by-nilpotent, where is generated by all nilpotent residuals of finitely generated subgroups. -- Further, the authors consider groups in which every subgroup is a subgroup of finite index of a subnormal subgroup.
[H.Heineken (Würzburg)
Groups with all subgroups -subnormal
No full textThe authors continue their investigation of -groups, i.e. the groups , in which every subgroup is -subnormal (there exists an -series from to , that is a finite series such that is finite or is normal in ).\par The main results of the article are the following theorems: Theorem 1. A torsion -groups is an extension of a finite group by a group in which all subgroups are subnormal. Theorem 2. The primary components of the subgroup generated by the nilpotent residuals of the finitely generated subgroups of an -group are finite.
[Igor Subbotin (Los Angeles)
Normal subgroups in the subgroup lattices of finite -groups
No full textGiven a finite -group we construct inductively a chief series only by lattice theoretical means. This improves a result of R. Schmidt that shows that any projectivity between finite groups and sends a chief series of into a chief series of
A CHARACTERIZATION OF HN: ADDENDUM
No full textIn the paper mentioned in the title [{\it C. Franchi, M. Mainardis} and {\it R. Solomon}, J. Group Theory 11, No. 3, 357-369 (2008; Zbl 1151.20014)] the Harada-Norton sporadic simple group was characterized as a group of bicharacteristic . In this note we extend that result to bicharacteristic for any prime greater than . Our result follows from the fact that, for such primes, a -local configuration of the type found in the Harada-Norton group for can appear in a finite group only when is equal to or
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