scholarly journals Semistability at ∞ of finitely generated groups, and solvable groups

1986 ◽  
Vol 24 (1-3) ◽  
pp. 259-269 ◽  
Author(s):  
Michael Mihalik
Keyword(s):  
Solvable Groups ◽  
Finitely Generated ◽  
Finitely Generated Groups

Characterization of finitely generated groups by types

2018 ◽  
Vol 28 (08) ◽  
pp. 1613-1632 ◽  
Author(s):  
A. G. Myasnikov ◽  
N. S. Romanovskii
Keyword(s):  
Free Surface ◽  
Nilpotent Group ◽  
Solvable Groups ◽  
Polycyclic Group ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Burnside Groups ◽  
Text Types ◽  
Rigid Group

In this paper we show that all finitely generated nilpotent, metabelian, polycyclic, and rigid (hence free solvable) groups [Formula: see text] are fully characterized in the class of all groups by the set [Formula: see text] of types realized in [Formula: see text]. Furthermore, it turns out that these groups [Formula: see text] are fully characterized already by some particular rather restricted fragments of the types from [Formula: see text]. In particular, every finitely generated nilpotent group is completely defined by its [Formula: see text]-types, while a finitely generated rigid group is completely defined by its [Formula: see text]-types, and a finitely generated metabelian or polycyclic group is completely defined by its [Formula: see text]-types. We have similar results for some non-solvable groups: free, surface, and free Burnside groups, though they mostly serve as illustrations of general techniques or provide some counterexamples.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups

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

2017 ◽  
Vol 20 (4) ◽  
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.

The word problem for semigroups satisfying x3 = x

1978 ◽  
Vol 84 (1) ◽  
pp. 11-19 ◽  
Author(s):  
J. A. Gerhard
Keyword(s):  
Word Problem ◽  
Upper Bound ◽  
Finitely Generated ◽  
Finitely Generated Groups

In the paper (4) of Green and Rees it was established that the finiteness of finitely generated semigroups satisfying xr = x is equivalent to the finiteness of finitely generated groups satisfying xr−1 = 1 (Burnside's Problem). A group satisfying x2 = 1 is abelian and if it is generated by n elements, it has at most 2n elements. The free finitely generated semigroups satisfying x3 = x are thus established to be finite, and in fact the connexion with the corresponding problem for groups can be used to give an upper bound on the size of these semigroups. This is a long way from an algorithm for a solution of the word problem however, and providing such an algorithm is the purpose of the present paper. The case x = x3 is of interest since the corresponding result for x = x2 was done by Green and Rees (4) and independently by McLean(6).

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