A homological characterization of locally finite groups

AbstractLet L = HK be a semidirect product of a normal locally finite π′-group H by a locally finite π′-group K, where π, is a set of primes. Suppose CK(H) = 1 and L is Sylow π-sparse (which in the countable case just says that the Sylow π-subgroups of L are conjugate). This paper completes the characterization of those groups which can occur as K—this had previously been obtained under the assumption that L is locally soluble. The answer is the same—essentially that the groups occurring are those having a subgroup of finite index which is a subdirect product of so-called “pinched” groups.

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Locally Finite Groups

10.1016/s0924-6509(08)x7028-x ◽

1973 ◽

Keyword(s):

Finite Groups ◽

Locally Finite ◽

Locally Finite Groups

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Examples of theories of presheaf type

10.1093/oso/9780198758914.003.0011 ◽

2018 ◽

Author(s):

Olivia Caramello

Keyword(s):

Finite Groups ◽

Linear Independence ◽

Geometric Theory ◽

Vector Spaces ◽

Linear Orders ◽

Locally Finite ◽

Strong Unit ◽

Locally Finite Groups ◽

Simplicial Sets ◽

Separable Extensions

This chapter discusses several classical as well as new examples of theories of presheaf type from the perspective of the theory developed in the previous chapters. The known examples of theories of presheaf type that are revisited in the course of the chapter include the theory of intervals (classified by the topos of simplicial sets), the theory of linear orders, the theory of Diers fields, the theory of abstract circles (classified by the topos of cyclic sets) and the geometric theory of finite sets. The new examples include the theory of algebraic (or separable) extensions of a given field, the theory of locally finite groups, the theory of vector spaces with linear independence predicates and the theory of lattice-ordered abelian groups with strong unit.

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Uncountable universal locally finite groups

Journal of Algebra ◽

10.1016/0021-8693(76)90150-2 ◽

1976 ◽

Vol 43 (1) ◽

pp. 168-175 ◽

Cited By ~ 25

Author(s):

Angus Macintyre ◽

Saharon Shelah

Keyword(s):

Finite Groups ◽

Locally Finite ◽

Locally Finite Groups

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Some properties of the growth and of the algebraic entropy of group endomorphisms

Journal of Group Theory ◽

10.1515/jgth-2016-0050 ◽

2017 ◽

Vol 20 (4) ◽

Cited By ~ 5

Author(s):

Anna Giordano Bruno ◽

Pablo Spiga

Keyword(s):

Finite Groups ◽

Addition Theorem ◽

Finitely Generated ◽

Growth Type ◽

Locally Finite ◽

Locally Finite Groups ◽

Algebraic Entropy ◽

Finitely Generated Groups ◽

Identity Automorphism ◽

Inner Automorphisms

AbstractWe study the growth of group endomorphisms, a generalization of the classical notion of growth of finitely generated groups, which is strictly related to algebraic entropy. We prove that the inner automorphisms of a group have the same growth type and the same algebraic entropy as the identity automorphism. Moreover, we show that endomorphisms of locally finite groups cannot have intermediate growth. We also find an example showing that the Addition Theorem for algebraic entropy does not hold for endomorphisms of arbitrary groups.

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On infinite groups in which all abelian subgroups are locally cyclic

Monatshefte für Mathematik ◽

10.1007/s00605-021-01594-w ◽

2021 ◽

Author(s):

Costantino Delizia ◽

Chiara Nicotera

Keyword(s):

Finite Groups ◽

Abelian Subgroup ◽

Periodic Case ◽

Infinite Groups ◽

Locally Finite ◽

Locally Finite Groups ◽

Periodic Groups ◽

Abelian Subgroups

AbstractThe structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

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