The number of configurations in the full shift with a given least period
aa r X i v : . [ m a t h . G R ] F e b The number of configurations in the full shift with a given leastperiod
Alonso Castillo-Ramirez ∗ and Miguel S´anchez- ´Alvarez Department of Mathematics, University Centre of Exact Sciences and Engineering,University of Guadalajara.
February 19, 2021
Abstract
For any group G and set A , consider the shift action of G on the full shift A G . A con-figuration x ∈ A G has least period H ≤ G if the stabiliser of x is precisely H . Amongother things, the number of such configurations is interesting as it provides an upper boundfor the size of the corresponding Aut( A G )-orbit. In this paper we show that if G is finitelygenerated and H is of finite index, then the number of configurations in A G with least period H may be computed using the M¨obius function of the lattice of subgroups of finite index in G . Moreover, when H is a normal subgroup, we classify all situations such that the numberof G -orbits with least period H is at most 10. Keywords:
Full shift; periodic configurations; subgroup lattice; M¨obius function.
MSC2020 codes:
Let A be a finite set and G a group. Consider the set A G of all functions from G to A equippedwith the shift action of G , defined for any g ∈ G and x ∈ A G as g · x ( h ) := x ( g − h ) , ∀ h ∈ G. Although we shall not focus on this, the set A G is usually seen as a topological space with theproduct topology of the discrete topology on A .The G -space A G is a fundamental object in areas such as symbolic dynamics and the theoryof cellular automata (e.g. see [2, 7]). Following [2], we call the elements of A G configurations .For any x ∈ A G , denote by G x the stabiliser of x in G , and by Gx the G -orbit of x .For a subgroup H of G , a configuration x ∈ A G has period H , or is H -periodic , if h · x = x for all h ∈ H , or, equivalently, if H ≤ G x . Denote by Fix( H ) the subset of A G consisting of all H -periodic configurations. It is known (see [2, Proposition 1.3.3]) that Fix( H ) is in bijectionwith A H \ G , where H \ G = { Hg : g ∈ G } is the set of rights cosets of H in G . Hence, if theindex [ G : H ] of H in G is finite, it follows that | Fix( H ) | = q [ G : H ] , where | A | = q . In particular,the configurations whose period is the trivial subgroup of G are known as aperiodic points , andhave been used in [4] as powerful tools to study the dynamics in A G and its subshifts , or subflows (i.e. closed G -equivariant subsets of A G ). ∗ Email: [email protected]
1e say that x ∈ A G has least period , or fundamental period , H if G x = H (c.f. [7, Definition1.1.3.]). In this paper we are interested in the number ψ H ( G ; A ) of configurations with leastperiod H : ψ H ( G ; A ) := |{ x ∈ A G : G x = H }| . If x, y ∈ A G satisfy that y = g · x , then G y = gG x g − ; hence, it is sometimes convenient toconsider the G -invariant set { x ∈ A G : [ G x ] = [ H ] } , where [ H ] := { gHg − : g ∈ G } is theconjugacy class of H , and its cardinality ψ [ H ] ( G ; A ) := |{ x ∈ A G : [ G x ] = [ H ] }| , As ψ H ( G ; A ) = ψ gHg − ( G ; A ) for all g ∈ G , we have ψ [ H ] ( G ; A ) = | [ H ] | ψ H ( G ; A ) . Finally, we also consider the number of G -orbits whose stabiliser is conjutate to H : α [ H ] ( G ; A ) := |{ Gx : [ G x ] = [ H ] }| . By the Orbit-Stabiliser Theorem, all G -orbits inside { x ∈ A G : [ G x ] = [ H ] } have size [ G : H ];therefore, we have α [ H ] ( G ; A ) [ G : H ] = ψ [ H ] ( G ; A ) . Besides being interesting for their own right, the above numbers have connections with thestructure of the automorphism group of A G . Recall that a map τ : A G → A G is G -equivariant if τ ( g · x ) = g · τ ( x ), for all g ∈ G , x ∈ A G . Let Aut( A G ) the group of all G -equivarianthomeomorphisms of A G . By Curtis-Heldund Theorem ([2, Theorem 1.8.1]), Aut( A G ) is thesame as the group of invertible cellular automata of A G . It follows by G -equivariance that forevery τ ∈ Aut( A G ), x ∈ A G , G x = G τ ( x ) . Thus, ψ G x ( G ; A ) is an upper bound for the cardinality of the Aut( A G )-orbit of x . Moreover, ifthe group G is finite, the structure of Aut( A G ) was described in [1, Theorem 3] asAut( A G ) ∼ = r Y i =1 (( N G ( H i ) /H i ) ≀ Sym α i ) , (1)where [ H ] , . . . , [ H r ] is the list of all different conjugacy classes of subgroups of G , and α i = α [ H i ] ( G ; A ), as defined above. Hence, the structure of Aut( A G ) completely depends on thequotient groups N G ( H i ) /H i , which may be easily calculated by knowing the group G , and theintegers α [ H i ] ( G ; A ), which depend on ψ H ( G ; A ).As ψ H ( G ; A ) is finite if and only if [ G : H ] is finite (see Lemma 1 below), we shall focus onfinite index subgroups of G . In the first part of this paper, we prove that, when G is finitelygenerated, the poset L ( G ) of finite index subgroups of G is a locally finite lattice, so we useM¨obius inversion to show that ψ H ( G ; A ) = X H ≤ K ≤ G µ ( H, K ) q [ G : K ] , (2)where µ is the M¨obius function of L ( G ). In the second part of this paper, we note that when H is a normal subgroup, then ψ H ( G ; A ) = ψ ( G/H ; A ) and α [ H ] ( G ; A ) = α [1] ( G/H ; A ). Hence,by computing the M¨obius function for all finite groups of size up to 7, we classify under whichsituations we have α [ H ] ( G ; A ) ≤ G = C n is a cyclicgroup and H = 1 is the trivial subgroup, α [1] ( C n ; A ) is equivalent to the number of aperiodicnecklaces of length n , and equation (2) gives the so-called Moreau’s necklace-counting function[9]. Moreover, α [1] ( C n ; A ) is also equivalent to the number of Lyndon words of length n (seeSec. 5.1. in [8]). For a finite group G , this equation may be derived using the result of Sec. 4in [6]. However, as far as we know, equation (2) had not been derived when G is an arbitraryfinitely generated group. 2 Periodic configurations when G is finitely generated For the rest of the paper, let A be a set with at least two elements and assume that { , } ⊆ A .We begin by justifying our claim that ψ H ( G ; A ) is finite if and only if [ G : H ] is finite. Lemma 1.
Let G be a group and H a subgroup of G . Then ψ H ( G ; A ) is finite if and only if [ G : H ] is finite.Proof. If [ G : H ] is finite, then ψ H ( G ; A ) is clearly finite, as every configuration with least period H is contained in Fix( H ) and | Fix( H ) | = | A | [ G : H ] < ∞ .Conversely, suppose that [ G : H ] is infinite. For each s ∈ G , consider the configuration x s ∈ A G defined by x s ( g ) = ( g ∈ Hs h ∈ H , then h · x s ( g ) = x s ( h − g ) = x s ( g ), as h − g ∈ Hs if and only if g ∈ Hs . Hence, H ≤ G x s . On the other hand, if k · x s ( g ) = x s ( k − g ) = x s ( g ) for every g ∈ G , in particular wehave x s ( k − s ) = x s ( s ) = 1, which implies that k − s ∈ Hs . Therefore, k ∈ H , which shows that G x s = H . As [ G : H ] is infinite, we have constructed infinitely many configurations with leastperiod H , which establishes that ψ H ( G ; A ) is infinite.We shall recall some basic definitions on posets; for further details see [12, Ch. 3]. Let ( P, ≤ )be a poset. Given s, t ∈ P with s ≤ t , define the closed interval [ s, t ] := { u ∈ P : s ≤ u ≤ t } . Wesay that P is locally finite if every closed interval of P is finite. A chain of P is a subposet S of P that is totally ordered, i.e. any two elements of S are comparable. For t ∈ P , the principalorder ideal generated by t is Λ t := { s ∈ P : s ≤ t } , and the principal dual order ideal generatedby t is V t := { s ∈ P : s ≥ t } .A lattice is a poset ( L, ≤ ) for which every pair of elements s, t ∈ L has a lest upper bound,denoted by s ∨ t and read s join t , and a greatest lower bound, denoted by s ∧ t and read s meet t . The M¨obius function of a locally finite poset ( P, ≤ ) is a map µ : P × P → Z definedinductively by the following equations: µ ( a, a ) = 1 , ∀ a ∈ P,µ ( a, b ) = 0 , ∀ a b, X a ≤ c ≤ b µ ( a, c ) = 0 , ∀ a < b. The M¨obius function is the inverse of the zeta function of a locally finite poset, and itimportantly satisfies the so-called M¨obius inversion formula (see [12, Sec. 3.7]). In this sectionwe shall use the dual form of the M¨obius inversion formula [12, Proposition 3.7.2].
Theorem 1 (M¨obius inversion formula, dual form) . Let P be a poset for which every principaldual order ideal V t is finite. Consider functions f, g : P → K , where K is a field. Then g ( t ) = X s ≥ t f ( s ) , ∀ t ∈ P, if and only if f ( t ) = X s ≥ t g ( s ) µ ( t, s ) , ∀ t ∈ P. For any group G , it is standard to consider the poset of all subgroups of G ordered byinclusion. Here, we shall consider the poset L ( G ) of all subgroups of G of finite index orderedby inclusion. The following is a key observation for this section.3 emma 2. The poset L ( G ) is a lattice. Furthermore, if G is finitely generated, then for every H ∈ L ( G ) , the principal dual order ideal V H = { K ≤ G : H ≤ K } is finite, so L ( G ) is a locallyfinite lattice.Proof. We shall show that L ( G ) is a sublattice of the subgroups lattice of G by showing that itis closed under the join, given by H ∨ J = h H ∪ J i , and the meet, given by H ∧ J = H ∩ J .Let H and K be subgroups of G such that H ≤ K . By [11, Theorem 3.1.3], the indicessatisfy, as cardinal numbers, that [ G : H ] = [ G : K ][ K : H ] . Hence, if [ G : H ] is finite, then [ G : K ] must be finite. This implies that for any H, J ∈ L ( G ),then h H ∪ J i ∈ L ( G ). On the other hand, [11, Theorem 3.1.6] implies that H ∩ J ∈ L ( G ), andthe first part of the lemma follows.For the second part, for any H ∈ L ( G ) and K ∈ V H , the index of K in G must be a divisorof [ G : H ]. The result follows as in a finitely generated group there are only finitely manysubgroups of a given finite index (see [11, Theorem 4.20]).The previous lemma allow us to use the M¨obius inversion formula for the poset L ( G ) when G is finitely generated. Let µ be the M¨obius function of L ( G ). Remark 1.
Let ˆ L ( G ) be the lattice of all subgroups of G . The proof of Lemma 2 implies thatclosed intervals in L ( G ) coincide with closed intervals in ˆ L ( G ). Note that for any H, J ∈ L ( G ),the value µ ( H, J ) only depends on the poset [
H, J ]. Therefore, for every
H, J ∈ L ( G ), µ ( H, J ) = ˆ µ ( H, J ) , where ˆ µ is the M¨obius function of ˆ L ( G ). Theorem 2.
Let G be a finitely generated group and let H be a subgroup of G of finite index,and A be a finite set of size q . Then, ψ H ( G ; A ) = X H ≤ K ≤ G µ ( H, K ) q [ G : K ] . Proof.
It follows from the definitions that | Fix( H ) | = X K ≥ H ψ K ( G ; A ) . By Lemma 2 this summation is finite and we may use Theorem 1, with g ( H ) = | Fix( H ) | and f ( K ) = ψ K ( G ; A ). Therefore, we obtain ψ H ( G ; A ) = X K ≥ H µ ( H, K ) | Fix( K ) | . The result follows as | Fix( K ) | = q [ G : K ] by [2, Proposition 1.3.3]. Corollary 1.
Let G be a finitely generated group and let H be a subgroup of G of finite index,and A a finite set of size q . Suppose that the interval from H to G consists of a chain H = H < H < · · · < H k = G . Then, ψ H ( G ; A ) = q [ G : H ] − q [ G : H ] . In particular, if H is a maximal subgroup of G , then ψ H ( G ; A ) = q [ G : H ] − q. roof. By Theorem 2, ψ H ( G ; A ) = k X i =0 µ ( H, H i ) q [ G : H i ] . Now, by the definition of the M¨obius function, µ ( H, H ) = 1 ,µ ( H, H ) = − ,µ ( H, H i ) = 0 , ∀ i = 2 , , . . . , k. The result follows.
Corollary 2.
Let G be a finitely generated group and let H be a subgroup of G of finite index,and A a finite set of size q . Then, ψ [ H ] ( G ; A ) = | [ H ] | X H ≤ K ≤ G µ ( H, K ) q [ G : K ] ,α [ H ] ( G ; A ) = | [ H ] | [ G : H ] X H ≤ K ≤ G µ ( H, K ) q [ G : K ] . In this section we shall specialise on the case when H is a normal subgroup of G of finite index.In this case, the conjugacy class of H just contains H itself, so ψ H ( G ; A ) = ψ [ H ] ( G ; A ) . Denote by 1 the trivial subgroup. The following result has been noted in [1, Lemma 6].
Lemma 3.
Let G be any group and H a normal subgroup of G of finite index. Then, ψ H ( G ; A ) = ψ ( G/H ; A ) and α [ H ] ( G ; A ) = α [1] ( G/H ; A ) . Proof.
By [2, Proposition 1.3.7.], there is a
G/H -equivariant bijection between A G/H andFix( H ). Hence, configurations in A G/H with trivial stabiliser are in bijection to configurationsin A G with stabiliser H .The previous lemma allow us to use theory of the M¨obius function of the subgroup latticeof a finite group, which has been thoroughly studied for a variety of finite groups (e.g. see[3, 5, 10]).Using Lemma 3, the following result gives the values of ψ H ( G ; A ) in some particular caseswhen H is a normal subgroup of G . Lemma 4.
Let G be a finitely generated group and H a normal subgroup of G of finite index.Let n ∈ N , and p and p ′ be distinct primes.1. If G/H ∼ = Z n , then ψ H ( G ; A ) = P d | n ˜ µ ( d ) q n/d , where ˜ µ is the classical M¨obius functionof the poset ( N , | ) given by ˜ µ ( d ) = if d has a squared prime factor if d is square-free with an even number of prime factors − if d is square-free with an odd number of prime factors . . If G/H ∼ = Z p k , then ψ H ( G ; A ) = q p k − q p k − .3. If G/H ∼ = Z pp ′ , then ψ H ( G ; A ) = q pp ′ − q p − q p ′ + q .4. If G/H ∼ = Z p ⊕ Z p , for a prime p , then ψ H ( G ; A ) = q p − ( p + 1) q p + pq .Proof. Parts (1), (2) and (3) follows as it is well-known that µ (1 , Z n ) = ˜ µ ( n ) (as the subgrouplattice of Z n is isomorphic to the divisibility lattice of n ). For part (4), just observe thatgroup Z p ⊕ Z p has p + 1 subgroups isomophic to Z p , which account for all its proper nontrivialsubgroups.In the rest of this section, we shall focus on classifying when small values for α [ H ] ( G ; A )occur. The inspiration for this question is Lemma 5 in [1], which established, without using theM¨obius function, that α [ H ] ( G ; A ) = 1 if and only if [ G : H ] = 2 and | A | = 2. In general, theclassification of small values for α [ H ] ( G ; A ) is relevant as it classifies configurations with smallAut( A G )-orbits, and, when G is finite, it classifies the small degrees of the symmetric groupsappearing in the decomposition (1) of Aut( A G ).For x ∈ A G , we have G x = G if and only if x is a constant configuration. As we haveprecisely | A | constant configurations in A G , then α [ G ] ( G ; A ) = | A | . Hence, we shall exclude thecase H = G in the following theorem. PPPPPPPPP
G/H | A | Z Z Z Z Z S Z Z
18 312 2340 11160Table 1: Small values for α [ H ] ( G ; A ) with H normal in G . Theorem 3.
Let G be a finitely generated group, H be a proper normal subgroup of G of finiteindex, and A a finite set of size q ≥ .1. α [ H ] ( G ; A ) = 1 if and only if q = 2 and [ G : H ] = 2 .2. α [ H ] ( G ; A ) = 2 if and only if q = 2 and [ G : H ] = 3 , or q = 2 and G/H ∼ = Z ⊕ Z .3. α [ H ] ( G ; A ) = 3 if and only if q = 3 and [ G : H ] = 2 , or q = 2 and G/H ∼ = Z .4. α [ H ] ( G ; A ) = 6 if and only if q = 2 and [ G : H ] = 5 , or q = 4 and [ G : H ] = 2 .5. α [ H ] ( G ; A ) = 7 if and only if q = 2 and G/H ∼ = S . . α [ H ] ( G ; A ) = 8 if and only if q = 3 and [ G : H ] = 3 .7. α [ H ] ( G ; A ) = 9 if and only if q = 2 and G/H ∼ = Z .8. α [ H ] ( G ; A ) = 10 if and only if q = 5 and [ G : H ] = 2 .9. α [ H ] ( G ; A ) = 4 and α [ H ] ( G ; A ) = 5 for all q .Proof. By Corollary 1.7.2 in [4], q [ G : H ] − q [ G : H ] − ≤ α [1] ( G/H, A ) = α [ H ] ( G ; A ) . (This lower bound has been improved in Theorem 5 in [1], but the above is enough for thisproof). Hence, we see that α [1] ( G/H, A ) is a strictly increasing function on both [ G : H ] and q = | A | . Table 1 shows all values of α [1] ( G/H, A ) with [ G : H ] ≤ q ≤
5. Most ofthese values may be calculated by using the formulae of Lemma 4; the only exeption is the case
G/H ∼ = S , which was direcly computed using the M¨obius function of the subgroup latice of S .The result follows by inspection of Table 1. References [1] Castillo-Ramirez, A., Gadouleaum, M.: Cellular automata and finite groups. Nat. Comput. (2019) 445–458.[2] Ceccherini-Silberstein, T., Coornaert, M.: Cellular Automata and Groups. Springer Mono-graphs in Mathematics, Springer-Verlag Berlin Heidelberg (2010).[3] Dalla Volta, F., Zini, G.: On two M¨obius functions for a finite non-solvable group.arXiv:2004.02694 (2020).[4] Gao, S., Jackson, S., Seward, B.: Group Colorings and Bernoulli Subflows. Mem. Am. Math.Soc. , no. 1141, 1–239 (2016).[5] Hawkes, T., Isaacs, I.M., ¨Ozaydin, M.: On the M¨obius Function of a Finite Group. RockyMt. J. Math. , no. 4, 1003–1034 (1989).[6] Kerber, A.: Applied Finite Group Actions, 2nd ed. Algorithms and Combinatorics ,Springer, 1999.[7] Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge Uni-versity Press, 1995.[8] Lothaire, M.: Combinatorics on words, Cambridge University Press, 1997.[9] Moreau, C.: Sur les permutations circulaires distinctes (On distinct circular permutations),Nouv. Ann. Math. (1872) 309–331.[10] Pahlings, H.: On the M¨obius function of a finite group. Arch. Math.60