On the minimal number of generators of endomorphism monoids of full shifts Alonso Castillo-Ramirez∗ Department of Mathematics, University Centre of Exact Sciences and Engineering, University of Guadalajara, Guadalajara, Mexico. October 22, 2019 Abstract For a group G and a finite set A, denote by End(AG) the monoid of all continuous shift com- muting self-maps of AG and by Aut(AG) its group of units. We study the minimal cardinality of a generating set, known as the rank, of End(AG) and Aut(AG). In the first part, when G is a finite group, we give upper and lower bounds for the rank of Aut(AG) in terms of the number of conjugacy classes of subgroups of G. In the second part, we apply our bounds to show that if G has an infinite descending chain of normal subgroups of finite index, then End(AG) is not finitely generated; such is the case for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups. Keywords: Full shift, endomorphisms, automorphisms, cellular automata, minimal number of generators. MSC 2010: 37B10, 68Q80, 05E18, 20M20. 1 Introduction Let G be a group and A a finite set. The full shift AG is the set of all maps x : G → A, equipped with the shift action of G on AG: arXiv:1910.01081v2 [math.GR] 19 Oct 2019 (g · x)(h)= x(g−1h), ∀g, h ∈ G, x ∈ AG. We endow AG with the prodiscrete topology, which is the product topology of the discrete topology on A. An endomorphism of AG is a continuous shift commuting self-map of AG. These are fundamental objects in symbolic dynamics, and by Curtis-Hedlund Theorem (see [9, Theorem 1.8.1]) a map τ : AG → AG is an endomorphism if and only if it is a cellular automaton of AG (i.e. there is a finite S −1 subset S ⊆ G, called a memory set, and a function µ : A → A satisfying τ(x)(g)= µ((g · x)|S ), ∀x ∈ AG, g ∈ G). Equipped with composition of functions, the set End(AG) of endomorphisms of AG is a monoid. The group of units (i.e. group of invertible elements) of End(AG) is denoted by Aut(AG). When |A| ≥ 2 and G = Z, several interesting properties are known for Aut(AZ): it is a countable group that is not finitely generated and it contains an isomorphic copy of every finite group, as well as ∗Email: [email protected] 1 the free group on a countable number of generators (see [2] and [13, Sec. 13.2]). However, despite of several efforts, most of the algebraic properties of End(AG) and Aut(AG) still remain unknown. Given a subset T of a monoid M, the submonoid generated by T , denoted by hT i, is the smallest submonoid of M that contains T ; this is equivalent as defining hT i := {t1t2 ...tk ∈ M : ti ∈ T, ∀i, k ≥ 0}. We say that T is a generating set of M if M = hT i. The monoid M is said to be finitely generated if it has a finite generating set. The rank of M is the minimal cardinality of a generating set: Rank(M) := min{|T | : M = hT i}. The question of finding the rank of a monoid is important in semigroup theory; it has been answered for several kinds of transformation monoids and Rees matrix semigroups (e.g., see [1, 11]). For the case of monoids of endomorphisms of full shifts over finite groups, the question has been addressed in [6, 7, 8]; in particular, the rank of Aut(AG) when G is a finite cyclic group has been examined in detail in [5]. In this paper, we study the rank of Aut(AG) and End(AG). In Section 2, we introduce notation and review some basic facts on group theory and the Rank function. In Section 3, when G is a finite group, we use the structure theorem for Aut(AG) obtained in [7] to provide upper and lower bounds for the rank of Aut(AG) in terms of the number of conjugacy classes of subgroups of G. We specialize in some particular cases such as cyclic, dihedral, Dedekind, and permutation groups. Finally, in Section 4, we apply our bounds to provide an elementary proof of the following theorem. Theorem 1. Let A be a finite set, and let G be a group has an infinite descending chain of normal subgroups of finite index in G. Then, End(AG) is not finitely generated. This theorem implies that End(AG) is not finitely generated for wide classes of infinite groups, such as infinite residually finite or infinite locally graded groups. However, it does not cover the cases of infinite groups with few normal subgroups of finite index, such as infinite symmetric groups or infinite simple groups. This paper is an extended version of [8]. All sections have been restructured, the exposition has been improved, and the results of Section 4 have been greatly extended. Moreover, the technical proof of Theorem 4, about dihedral groups, has been omitted. 2 Basic Results We assume the reader has certain familiarity with basic concepts of group theory. Let G be a group and A a finite set. The stabiliser and G-orbit of a configuration x ∈ AG are defined, respectively, by Gx := {g ∈ G : g · x = x} and Gx := {g · x : g ∈ G}. Stabilisers are subgroups of G, while the set of G-orbits forms a partition of AG. −1 Two subgroups H1 and H2 of G are conjugate in G if there exists g ∈ G such that g H1g = H2. This defines an equivalence relation on the subgroups of G. Denote by [H] the conjugacy class of H ≤ G. A subgroup H ≤ G is normal if [H] = {H} (i.e. g−1Hg = H for all g ∈ G). Let −1 NG(H) := {g ∈ G : H = g Hg} ≤ G be the normaliser of H in G. Note that H is always a normal subgroup of NG(H). Denote by r(G) the total number of conjugacy classes of subgroups of G, and by ri(G) the number of conjugacy classes [H] such that H has index i in G: r(G) := |{[H] : H ≤ G}|, ri(G) := |{[H] : H ≤ G, [G : H]= i}|. 2 For any H ≤ G, denote G α[H](G; A) := |{Gx ⊆ A : [Gx] = [H]}|. This number may be calculated using the Mobius function of the subgroup lattice of G, as shown in [7, Sec. 4]. For any integer α ≥ 1, let Sα be the symmetric group of degree α. The wreath product of a group C by Sα is the set α C ≀ Sα := {(v; φ) : v ∈ C , φ ∈ Sα} φ α equipped with the operation (v; φ) · (w; ψ) = (v · w ; φψ), for any v, w ∈ C , φ, ψ ∈ Sα, where φ acts on w by permuting its coordinates: φ φ w = (w1, w2, . , wα) := (wφ(1), wφ(2), . , wφ(α)). In fact, as may be seen from the above definitions, C ≀Sα is equal to the external semidirect product α α α C ⋊ϕ Sα, where ϕ : Sα → Aut(C ) is the action of Sα of permuting the coordinates of C . For a more detailed description of the wreath product see [1]. The Rank function on monoids does not behave well when taking submonoids or subgroups: in other words, if N is a submonoid of M, there may be no relation between Rank(N) and Rank(M). For example, if M = Sn is the symmetric group of degree n ≥ 3 and N is a subgroup of Sn generated n by ⌊ 2 ⌋ commuting transpositions, then Rank(Sn) = 2, as Sn may be generated by a transposition n and an n-cycle, but Rank(N) = ⌊ 2 ⌋. It is even possible that M is finitely generated but N is not finitely generated (such as the case of the free group on two symbols and its commutator subgroup). However, there are some tools that we may use to bound the rank. For any subset U of a monoid M, the relative rank of U in M is Rank(M : U) = min{|W | : M = hU ∪ W i}. When M is a finite monoid and U is the group of units of M, we have the basic identity Rank(M) = Rank(M : U) + Rank(U), (1) which follows as any generating set for M must contain a generating set for U (see [1, Lemma 3.1]). The relative rank of Aut(AG) in End(AG) has been established in [7, Theorem 7] for finite Dedekind groups (i.e. groups in which all subgroups are normal). Lemma 1. For any finite group G and finite set A, Rank(Aut(AG)) ≤ Rank(End(AG)). Proof. As G and A are both finite, End(AG) is a finite monoid. The result follows by (1). Lemma 2. Let G and H be a groups, and let N be a normal subgroup of G. Then: 1. Rank(G/N) ≤ Rank(G). 2. Rank(G × H) ≤ Rank(G) + Rank(H). 3. Rank(G ≀ Sα) ≤ Rank(G) + Rank(Sα), for any α ≥ 1. 4. Rank(Zd ≀ Sα) = 2, for any d, α ≥ 2. Proof. Parts 1 and 2 are straightforward. For parts 3 and 4, see [7, Corollary 5] and [5, Lemma 5], respectively. 3 3 Finite groups The main tool of this section is the following structure theorem for Aut(AG). Theorem 2 ([7]). Let G be a finite group and A a finite set of size q ≥ 2. Let [H1],..., [Hr] be the list of all different conjugacy classes of subgroups of G.