Cellular Automata and Groups

Glossary

Groups:

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 1_G ∈ G (necessarily unique) such that, for all g in G, 1_Gg = g1_G = 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 = 1_G (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, s_i ∈ S and α_i ∈ ℤ ∖ {0} for 1 ≤ i ≤ n, and such that s_i ≠ s_i+1 for 1 ≤ i ≤ n − 1. Such a set S is called a free basis for F. The cardinality of Sis an invariant...