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 results on finite factorized 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
The well-known fact that a product of two normal supersoluble subgroups of a group is not necessarily supersoluble shows that the saturated formation of supersoluble groups need not be closed under the product of normal subgroups. This makes interesting the study of factorized groups whose subgroup factors are connected by certain permutability properties. Baer (see [2]) proved that if a group G is the product of two normal supersoluble subgroups, then G is supersoluble if and only if the commutator subgroup of G is nilpotent. This result has been generalized by Asaad and Shaalan ([1]) in the following sense: If G is the product of two subgroups H and K such that H permutes with every subgroup of K and K permutes with every subgroup of H, that is, G is the mutually permutable product of H and K, and G', the commutator subgroup of G is nilpotent, then G is supersoluble. Moreover they prove that in the case G = HK such that every subgroup of H permutes with every subgroup of K, that is, G is the totally permutable product of H and K, then if the factors H and K are supersoluble the group G is also supersoluble.
Further studies have been done by several authors within the framework of formation theory.
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- Groups St Andrews 2001 in Oxford , pp. 27 - 30Publisher: Cambridge University PressPrint publication year: 2003