Glossary
- Groups:
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A group is a set G endowed with a binary operation G × G ∋ (g, h) ↦ gh ∈ G, called the multiplication, that satisfies the following properties: (i) for all g, h and k in G, (gh)k = g(hk) (associativity); (ii) there exists an element 1G ∈ G (necessarily unique) such that, for all g in G, 1Gg = g1G = g (existence of the identity element); (iii) for each g in G, there exists an element g−1 ∈ G (necessarily unique) such that gg−1 = g−1g = 1G (existence of the inverses).
A group G is said to be Abelian (or commutative) if the operation is commutative, that is, for all g, h ∈ G one has gh = hg.
A group F is called free if there is a subset S ⊂ F such that any element g of F can be uniquely written as a reduced word on S, i.e. in the form \( g={s}_1^{\alpha_1}{s}_2^{\alpha_2}\cdots {s}_n^{\alpha_n} \), where n ≥ 0, si ∈ S and αi ∈ ℤ ∖ {0} for 1 ≤ i ≤ n, and such that si ≠ si+1 for 1 ≤ i ≤ n − 1. Such a set S is called a free basis for F. The cardinality of Sis an invariant...
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Ceccherini-Silberstein, T., Coornaert, M. (2009). Cellular Automata and Groups. In: Adamatzky, A. (eds) Cellular Automata. Encyclopedia of Complexity and Systems Science Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8700-9_52
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