Book contents
- Frontmatter
- Contents
- Contents of Volume 2
- Introduction
- Permutability and subnormality in finite groups
- (Pro)-finite and (topologically) locally finite groups with a CC-subgroup
- Table algebras generated by elements of small degrees
- Subgroups which are a union of a given number of conjugacy classes
- Some results on finite factorized groups
- On nilpotent-like Fitting formations
- Locally finite groups with min-p for all primes p
- Quasi-permutation representations of 2-groups of class 2 with cyclic centre
- Groups acting on bordered Klein surfaces with maximal symmetry
- Breaking points in subgroup lattices
- Group actions on graphs, maps and surfaces with maximum symmetry
- On dual pronormal subgroups and Fitting classes
- (p, q, r)-generations of the sporadic group O'N
- Computations with almost-crystallographic groups
- Random walks on groups: characters and geometry
- On distances of 2-groups and 3-groups
- Zeta functions of groups: the quest for order versus the flight from ennui
- Some factorizations involving hypercentrally embedded subgroups in finite soluble groups
- Elementary theory of groups
- Andrews-Curtis and Todd-Coxeter proof words
- Short balanced presentations of perfect groups
- Finite p-extensions of free pro-p groups
- Elements and groups of finite length
- Logged rewriting and identities among relators
- A characterization of F4(q) where q is an odd prime power
- On associated groups of rings
Some factorizations involving hypercentrally embedded subgroups in finite soluble groups
Published online by Cambridge University Press: 11 January 2010
- Frontmatter
- Contents
- Contents of Volume 2
- Introduction
- Permutability and subnormality in finite groups
- (Pro)-finite and (topologically) locally finite groups with a CC-subgroup
- Table algebras generated by elements of small degrees
- Subgroups which are a union of a given number of conjugacy classes
- Some results on finite factorized groups
- On nilpotent-like Fitting formations
- Locally finite groups with min-p for all primes p
- Quasi-permutation representations of 2-groups of class 2 with cyclic centre
- Groups acting on bordered Klein surfaces with maximal symmetry
- Breaking points in subgroup lattices
- Group actions on graphs, maps and surfaces with maximum symmetry
- On dual pronormal subgroups and Fitting classes
- (p, q, r)-generations of the sporadic group O'N
- Computations with almost-crystallographic groups
- Random walks on groups: characters and geometry
- On distances of 2-groups and 3-groups
- Zeta functions of groups: the quest for order versus the flight from ennui
- Some factorizations involving hypercentrally embedded subgroups in finite soluble groups
- Elementary theory of groups
- Andrews-Curtis and Todd-Coxeter proof words
- Short balanced presentations of perfect groups
- Finite p-extensions of free pro-p groups
- Elements and groups of finite length
- Logged rewriting and identities among relators
- A characterization of F4(q) where q is an odd prime power
- On associated groups of rings
Summary
Introduction
All groups considered in this note are finite. If H is a subgroup of a group G, we write and Recall that Subgroups which permute with all Sylow subgroups of the group, or S-permutable subgroups, were introduced by Kegel in his seminal paper [K 62]. P. Schmid, in [Sch 98], presented an extensive and elegant study of these subgroups. In that paper it is proved that for a core-free S-permutable subgroup T of a group G which also permutes with the normalizer of a Sylow subgroup N, then T ≤ N ([Sch 98]; Prop. D). Since the hypercenter Z∞(G) of the group G, i.e. the last member of the ascending central series of G, is the intersection of the normalizers of all Sylow subgroups of G, we have that for a subgroup T of G the following are equivalent:
(i) T is an S-permutable subgroup which permutes with the normalizers of all Sylow subgroups of G, and
(ii) TG/TG ≤ Z∞(G/TG), for TG = ∩g∈GTg, the core of T in G.
Such a subgroup is said to be a hypercentrally embedded subgroup. We focus our attention on these special S-permutable subgroups. Hypercentrally embedded subgroups enjoy very good factorization properties. In fact, previously, Carocca and Maier, in [CM 98], had characterized hypercentrally embedded subgroups as those subgroups which permute with all pronormal subgroups. In this note we present some factorizations of hypercentrally embedded subgroups with some special types of subgroups which, in general, are not pronormal.
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- Groups St Andrews 2001 in Oxford , pp. 190 - 196Publisher: Cambridge University PressPrint publication year: 2003
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