Finitely generated groups

A finitely generated group is a subgroup of a given group (G, *) that is denoted by 〈S〉, where S is a subset of G. This subgroup consists of all elements of G that can be expressed as finite combinations (under the operator *) of points in S and their inverses. The term is often used in the context of topological entropy, where Ghys, Langevin, and Walczak proposed a definition for finitely generated groups and pseudogroups of continuous transformations.From: Computational Topology for Biomedical Image and Data Analysis [2019], Some properties on topological entropy of free semigroup action [2018]

Elements of topology and homology

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Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019

Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama

Finitely generated group Given a group (G, *) and a subset S ⊂ G, the finitely generated group S denoted by 〈S〉 is a subgroup consisting of all elements of G that can be expressed as finite combinations (under the operator *) of points in S and their inverses.

Symmetries and Group Theory

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Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018

Mattias Blennow

The above example shows us a finitely generated group that has an infinite number of elements. However, some groups contain only a finite number of elements and are therefore called finite groups. An advantage of finite groups is the possibility of constructing a group table (or Cayley table) of the form:

The Garden of Eden theorem for cellular automata on group sets

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Published in International Journal of Parallel, Emergent and Distributed Systems, 2019

Simon Wacker

[Finitely] Let G be a finitely generated group, let M be the vertices of a Cayley graph of G, and let be the left group action of G on M by left multiplication. The cell space is finitely and symmetrically right generated and hence right tractable.

Finitely generated groups

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Elements of topology and homology

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The Garden of Eden theorem for cellular automata on group sets