Bounding the minimal number of generators of groups and monoids of cellular automata
aa r X i v : . [ m a t h . G R ] J un Bounding the minimal number of generators of groups and monoidsof cellular automata
Alonso Castillo-Ramirez ∗ and Miguel Sanchez-Alvarez † Department of Mathematics, University Centre of Exact Sciences and Engineering,University of Guadalajara, Guadalajara, Mexico.
June 11, 2019
Abstract
For a group G and a finite set A , denote by CA( G ; A ) the monoid of all cellular automataover A G and by ICA( G ; A ) its group of units. We study the minimal cardinality of a generatingset, known as the rank , of ICA( G ; A ). In the first part, when G is a finite group, we give upperbounds for the rank in terms of the number of conjugacy classes of subgroups of G . The casewhen G is a finite cyclic group has been studied before, so here we focus on the cases when G is a finite dihedral group or a finite Dedekind group. In the second part, we find a basic lowerbound for the rank of ICA( G ; A ) when G is a finite group, and we apply this to show that,for any infinite abelian group H , the monoid CA( H ; A ) is not finitely generated. The same istrue for various kinds of infinite groups, so we ask if there exists an infinite group H such thatCA( H ; A ) is finitely generated. Keywords:
Monoid of cellular automata, invertible cellular automata, minimal number ofgenerators.
MSC 2010:
The theory of cellular automata (CA) has important connections with many areas of mathematics,such as group theory, topology, symbolic dynamics, coding theory, and cryptography. Recently, in[5, 6, 7], links with semigroup theory have been explored, and, in particular, questions have beenconsidered on the structure of the monoid of all CA and the group of all invertible CA over a givenconfiguration space. The goal of this paper is to bound the minimal number of generators, knownin semigroup theory as the rank , of groups of invertible CA.Let G be a group and A a finite set. Denote by A G the configuration space , i.e. the set of allfunctions of the form x : G → A . The shift action of G on A G is defined by g · x ( h ) = x ( g − h ), forall x ∈ A G , g, h ∈ G . We endow A G with the prodiscrete topology , which is the product topology ofthe discrete topology on A . A cellular automaton over A G is a transformation τ : A G → A G suchthat there is a finite subset S ⊆ G and a function µ : A S → A satisfying τ ( x )( g ) = µ (( g − · x ) | S ) , ∀ x ∈ A G , g ∈ G, ∗ Email: [email protected] (corresponding author) † Email: miguel [email protected] | S denotes the restriction to S of a configuration in A G .Curtis-Hedlund Theorem ([8, Theorem 1.8.1]) establishes that a function τ : A G → A G is acellular automaton if and only if it is continuous in the prodiscrete topology of A G and commuteswith the shift action (i.e. τ ( g · x ) = g · τ ( x ), for all x ∈ A G , g ∈ G ). By [8, Corollary 1.4.11],the composition of two cellular automata over A G is a cellular automaton over A G . This impliesthat, equipped with composition of functions, the set CA( G ; A ) of all cellular automata over A G is amonoid. The group of units (i.e. group of invertible elements) of CA( G ; A ) is denoted by ICA( G ; A ).When | A | ≥ G = Z , many interesting properties are known for ICA( Z ; A ): for example, everyfinite group, as well as the free group on a countable number of generators, may be embedded inICA( Z ; A ) (see [2]). However, despite of several efforts, most of the algebraic properties of CA( G ; A )and ICA( G ; A ) still remain unknown.Given a subset T of a monoid M , the submonoid generated by T , denoted by h T i , is thesmallest submonoid of M that contains T ; this is equivalent as defining h T i := { t t . . . t k ∈ M : t i ∈ T, ∀ i, k ≥ } . We say that T is a generating set of M if M = h T i . The monoid M is said tobe finitely generated if it has a finite generating set. The rank of M is the minimal cardinality of agenerating set: Rank( M ) := min {| T | : M = h T i} . The question of finding the rank of a monoid is important in semigroup theory; it has beenanswered for several kinds of transformation monoids and Rees matrix semigroups (e.g., see [1, 11]).For the case of monoids of cellular automata over finite groups, the question has been addressed in[6, 7]; in particular, the rank of ICA( G ; A ) when G is a finite cyclic group has been examined indetail in [5].For any subset U of a monoid M , the relative rank of U in M isRank( M : U ) = min {| W | : M = h U ∪ W i} . When U is the group of units of M , we have the basic identityRank( M ) = Rank( M : U ) + Rank( U ) , (1)which follows as the group U can only be generated by subsets of itself. The relative rank ofICA( G ; A ) in CA( G ; A ) has been established in [7, Theorem 7] for finite Dedekind groups (i.e.groups in which all subgroups are normal).In this paper, we study the rank of ICA( G ; A ) when G is a finite group. In Section 2, weintroduce notation and review some basic facts, including the structure theorem for ICA( G ; A )obtained in [7]. In Section 3, we give upper bounds for the rank of ICA( G ; A ), examining in detailthe cases when G is a finite dihedral group or a finite Dedekind group, but also obtaining someresults for a general finite group. In Section 4, we show that, when G is finite, the rank of ICA( G ; A )is at least the number of conjugacy classes of subgroups of G . As an application, we use this toprovide a simple proof that the monoid CA( G ; A ) is not finitely generated whenever G is an infiniteabelian group. This result implies that CA( G ; A ) is not finitely generated for various classes ofinfinite groups, such as free groups and the infinite dihedral group. Thus, we ask if there exists aninfinite group G such that the monoid CA( G ; A ) is finitely generated. We assume the reader has certain familiarity with basic concepts of group theory.2et G be a group and A a finite set. The stabiliser and G -orbit of a configuration x ∈ A G aredefined, respectively, by G x := { 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 A G .Two subgroups H and H of G are conjugate in G if there exists g ∈ G such that g − H g = H .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 − Hg = H for all g ∈ G ). Let N G ( H ) := { g ∈ G : H = g − Hg } ≤ G be the normaliser of H in G . Note that H is always anormal subgroup of N G ( H ). Denote by r ( G ) the total number of conjugacy classes of subgroups of G , and by r i ( G ) the number of conjugacy classes [ H ] such that H has index i in G : r ( G ) := |{ [ H ] : H ≤ G }| ,r i ( G ) := |{ [ H ] : H ≤ G, [ G : H ] = i }| . For any H ≤ G , denote α [ H ] ( G ; A ) := |{ Gx ⊆ A G : [ G x ] = [ H ] }| . This number may be calculated using the Mobius function of the subgroup lattice of G , as shownin [7, Sec. 4].For any integer α ≥
1, let S α be the symmetric group of degree α . The wreath product of agroup C by S α is the set C ≀ S α := { ( v ; φ ) : v ∈ C α , φ ∈ S α } equipped with the operation ( v ; φ ) · ( w ; ψ ) = ( v · w φ ; φψ ), for any v, w ∈ C α , φ, ψ ∈ S α , where φ actson w by permuting its coordinates: w φ = ( w , w , . . . , w α ) φ := ( w φ (1) , w φ (2) , . . . , w φ ( α ) ) . In fact, as may be seen from the above definitions, C ≀ S α is equal to the external semidirect product C α ⋊ ϕ S α , where ϕ : S α → Aut( C α ) is the action of S α of permuting the coordinates of C α . For amore detailed description of the wreath product see [1]. Theorem 1 ([7]) . Let G be a finite group and A a finite set of size q ≥ . Let [ H ] , . . . , [ H r ] be thelist of all different conjugacy classes of subgroups of G . Let α i := α [ H i ] ( G ; A ) . Then, ICA( G ; A ) ∼ = r Y i =1 (( N G ( H i ) /H i ) ≀ S α i ) . The Rank function on monoids does not behave well when taking submonoids or subgroups: inother words, if N is a submonoid of M , there may be no relation between Rank( N ) and Rank( M ).For example, if M = S n is the symmetric group of degree n ≥ N is a subgroup of S n generatedby ⌊ n ⌋ commuting transpositions, then Rank( S n ) = 2, as S n may be generated by a transpositionand an n -cycle, but Rank( N ) = ⌊ n ⌋ . It is even possible that M is finitely generated but N is notfinitely generated (such as the case of the free group on two symbols and its commutator subgroup).However, the following lemma gives us some elementary tools to bound the rank in some cases.3 emma 1. 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 ≀ S α ) ≤ Rank( G ) + Rank( S α ) , for any α ≥ .4. Rank( Z d ≀ S α ) = 2 , for any d, α ≥ . .Proof. Parts 1 and 2 are straightforward. For parts 3 and 4, see [7, Corollary 5] and [5, Lemma 5],respectively.We shall use Lemma 1 together with Theorem 1 in order to find upper bounds for ICA( G ; A ).Because of part 3 in Lemma 1, it is now relevant to determine some values of the α i ’s that appearin Theorem 1. Lemma 2.
Let G be a finite group and A a finite set of size q ≥ . Let H be a subgroup of G .1. α [ G ] ( G ; A ) = q .2. α [ H ] ( G ; A ) = 1 if and only if [ G : H ] = 2 and q = 2 .3. If q ≥ , then α [ H ] ( G ; A ) ≥ .Proof. Parts 1 and 2 correspond to Remark 1 and Lemma 5 in [7], respectively. For part 2, Supposethat q ≥ { , , } ⊆ A . Define configurations z , z , z ∈ A G as follows, z ( g ) = ( g ∈ H g H, z ( g ) = ( g ∈ H g H, z ( g ) = ( g ∈ H g H, All three configurations are in different orbits and G z i = H , for i = 1 , ,
3. Hence α [ H ] ( G ; A ) ≥ q ≥
3, our upper bounds cannot be refined by a more careful examination of thevalues of the α ′ i s , as, for all α ≥
3, we have Rank( S α ) = 2. In this section we investigate the rank of ICA( D n ; A ), where D n is the dihedral group of order2 n , with n ≥
1, and A is a finite set of size q ≥
2. We shall use the following standard presentationof D n : D n = (cid:10) ρ, s | ρ n = s = sρsρ = id (cid:11) . Lemma 3.
For any n ≥ and α ≥ , Rank( D n ≀ S α ) ≤ .Proof. By [5, Lemma 5], we know that h ρ i ≀ S α ∼ = Z n ≀ S α may be generated by two elements. Hence,by adding (( s, id , . . . , id); id) we may generate the whole D n ≀ S α with three elements.Given a subgroup H ≤ D n , we shall now analyze the quotient group N G ( H ) /H . Lemma 4.
Let G = D n and let H ≤ D n be a subgroup of odd index m . Then H is self-normalizing,i.e. N G ( H ) = H . roof. By [9, Theorem 3.3], all subgroups of D n with index m are conjugate to each other. By [9,Corollary 3.2], there are m subgroups of D n of index m , so | [ H ] | = m = [ D n : H ]. On the otherhand, by the Orbit-Stabilizer Theorem applied to the conjugation action of D n on its subgroupswe have | [ H ] | = [ D n : N G ( H )]. Therefore, [ D n : H ] = [ D n : N G ( H )] and N G ( H ) = H . Lemma 5.
Let G = D n and let H ≤ D n be a proper subgroup of even index m . Then H is normalin D n and D n /H ∼ = D m , except when n is even, m | n , and [ H ] = [ h ρ m , s i ] or [ H ] = [ h ρ m , ρs i ] , inwhich case N G ( H ) /H ∼ = Z .Proof. We shall use Corollary 3.2 and Theorem 3.3 in [9]. There are two cases to consider:1. Suppose n is odd. Then, D n has a unique subgroup of index m , so 1 = | [ H ] | = [ D n : N G ( H )].This implies that D n = N G ( H ), so H is normal in D n .2. Suppose that n is even. If m ∤ n , D n has a unique subgroup of index m , so H is normal bythe same argument as in the previous case. If m | n , then D n has m + 1 subgroups with index m partitioned into 3 conjugacy classes [ h ρ m/ i ], [ h ρ m , s i ] and [ h ρ m , rs i ] of sizes 1, m and m ,respectively. If | [ H ] | = 1, again H is normal. If | [ H ] | = m , then [ D n : N G ( H )] = [ D n : H ].Hence, [ N G ( H ) : H ] = 2, and N G ( H ) /H ∼ = Z .The fact that D n /H ∼ = D m whenever H is normal follows by [9, Theorem 2.3].Let d( n ) be number of divisors of n , including 1 and n itself. Let d − ( n ) and d + ( n ) be thenumber of odd and even divisors of n , respectively.When n is odd, D n has exactly 1 conjugacy class of subgroups of index m , for every m | n ;hence, if n is odd, r ( D n ) = d(2 n ). When n is even, D n has exactly 1 conjugacy class of index m ,when m | n is odd or m ∤ n , and exactly 3 conjugacy classes when m | n is even and m | n ; hence,if n is even, r ( D n ) = d(2 n ) + 2d + ( n ). Theorem 2.
Let n ≥ be an integer and A a finite set of size at least . Rank(ICA( D n ; A )) ≤ − (2 n ) + 3d + (2 n ) − if n is odd and q = 2 , − (2 n ) + 3d + (2 n ) − if n is odd and q ≥ , − (2 n ) + 3d + (2 n ) + 2d + ( n ) − if n is even and q = 2 , − (2 n ) + 3d + (2 n ) + 4d + ( n ) − if n is even and q ≥ .Proof. Let [ H ] , . . . , [ H r ] be the conjugacy classes of subgroups of D n . Let m i = [ G : H i ], with m r − = 2 and m r = 1 (i.e. H r = D n ). Define α i := α [ H i ] ( G ; A ).Let n be odd. Then, by Theorem 1, and Lemmas 4 and 5,ICA( D n ; A ) ∼ = Y m i | n odd S α i × Y 1) + 2= 2d − (2 n ) + 3d + (2 n ) − . q = 2, Lemma 2 shows that α r = 2 and α r − = 1, so S α r ∼ = S and Z ≀ S α r − ∼ = Z . Therefore,Rank(ICA( D n ; A )) ≤ X 1) + 3(d + (2 n ) − 1) + 2= 2d − (2 n ) + 3d + (2 n ) − . Now let n be even. Then,ICA( D n ; A ) ∼ = Y m i | n odd S α i × Y 4= 2d − (2 n ) + 3(d + (2 n ) − 1) + 2 + 4d + ( n )= 2d − (2 n ) + 3d + (2 n ) + 4d + ( n ) − . When q = 2, by Lemma 2 we have S α r ∼ = S , Z ≀ S α r − ∼ = Z and Z ≀ S α i ∼ = Z , soRank(ICA( D n ; A )) ≤ X 2= 2(d − (2 n ) − 1) + 3(d + (2 n ) − 1) + 2d + ( n ) + 2= 2d − (2 n ) + 3d + (2 n ) + 2d + ( n ) − . Example 1. Let A be a finite set of size q ≥ 2. By the previous theorem,Rank(ICA( D ; A )) ≤ ( − (6) + 3d + (6) − · · − q = 2 , d − (6) + 3 d + (6) − · · − q ≥ . On the other hand,Rank(ICA( D ; A )) ≤ ( − (8) + 3d + (8) + 2d + (4) − q = 2 , d − (8) + 3 d + (8) + 4 d + (4) − q ≥ . Recall that r ( G ) denotes the total number of conjugacy classes of subgroups of G and r i ( G ) thenumber of conjugacy classes [ H ] such that H has index i in G . The following results are animprovement of [7, Corollary 5]. Theorem 3. Let G be a finite Dedekind group and A a finite set of size q ≥ . Let r := r ( G ) and r i := r i ( G ) . Let p , . . . , p s be the prime divisors of | G | and define r P := P si =1 r p i . Then, Rank(ICA( G ; A )) ≤ ( ( r − r P − G ) + 2 r − r − , if q = 2 , ( r − r P − G ) + 2 r, if q ≥ . roof. Let H , H , . . . , H r be the list of different subgroups of G with H r = G . If H i is a subgroupof index p k , then ( G/H i ) ≀ S α i ∼ = Z p k ≀ S α i is a group with rank 2, by Lemma 1. Thus, by Theorem1 we have: Rank(ICA( G ; A )) ≤ r − X i =1 Rank(( G/H i ) ≀ S α i ) + Rank( S q ) ≤ X [ G : H i ]= p k X [ G : H i ] = p k (Rank( G ) + 2) + 2= 2 r P + ( r − r P − G ) + 2) + 2= ( r − r P − G ) + 2 r. If q = 2, we may improve this bound by using Lemma 2:Rank(ICA( G ; A )) ≤ X [ G : H i ]=2 Rank(( G/H i ) ≀ S ) + X [ G : H i ]= p k =2 Rank(( G/H i ) ≀ S α i )+ X =[ G : H i ] = p k Rank(( G/H i ) ≀ S α i ) + Rank( S )= r + 2( r P − r ) + ( r − r P − G ) + 2) + 1= ( r − r P − G ) + 2 r − r − . Example 2. The smallest example of a nonabelian Dedekind group is the quaternion group Q = h x, y | x = x y − = y − xyx = id i , which has order 8. It is generated by two elements, and it is noncyclic, so Rank( Q ) = 2. Moreover, r = r ( Q ) = 6 and, as 2 is the only prime divisor of 8, we have r P = r = 3. Therefore,Rank(ICA( Q ; A )) ≤ ( (6 − − · · − − , if q = 2 , (6 − − · · , if q ≥ . Corollary 1. Let G be a finite Dedekind group and A a finite set of size q ≥ . With the notationof Theorem 3, Rank(CA( G ; A )) ≤ ( ( r − r P − G ) + r ( r + 5) − r − , if q = 2( r − r P − G ) + r ( r + 5) , otherwise.Proof. The result follows by Theorem 3, identity (1) and the basic upper bound for the relativerank that follows from [7, Theorem 7]:Rank(CA( G ; A ) : ICA( G ; A )) ≤ ((cid:0) r (cid:1) + r − r if q = 2 (cid:0) r (cid:1) + r, otherwise.Now focus now when G is not necessarily a Dedekind group.7 emma 6. Let G be a finite group and H a subgroup of G of prime index p . Let A be a finite setof size q ≥ and α := α [ H ] ( G ; A ) . Then Rank (( N G ( H ) /H ) ≀ S α ) ≤ ( if p = 2 and q = 22 otherwise . Proof. By Lagrange’s theorem, N G ( H ) = H or N G ( H ) = G . Hence, in order to find an upperbound for the above rank, we assume that H is normal in G . As the index is prime, G/H ∼ = Z p . If p = 2 and q = 2, Lemma 2 shows that α = 1, so Rank( Z ≀ S ) = 1. For the rest of the cases wehave that Rank( Z p ≀ S α ) = 2, by Lemma 1.The length of G (see [3, Sec. 1.15]) is the length ℓ := ℓ ( G ) of the longest chain of propersubgroups 1 = G < G < · · · < G ℓ = G. The lengths of the symmetric groups are known by [4]: ℓ ( S n ) = ⌈ n/ ⌉ − b ( n ) − 1, where b ( n ) is thenumbers of ones in the base 2 expansion of n . As, ℓ ( G ) = ℓ ( N ) + ℓ ( G/N ) for any normal subgroup N of G , the length of a finite group is equal to the sum of the lengths of its compositions factors;hence, the question of calculating the length of all finite groups is reduced to calculating the lengthof all finite simple groups. Moreover, ℓ ( G ) ≤ log ( | G | ) (see [4, Lemma 2.2]). Lemma 7. Let G be a finite group and H a subgroup of G . Let A be a finite set of size q ≥ and α := α [ H ] ( G ; A ) . Then, Rank (( N G ( H ) /H ) ≀ S α ) ≤ ℓ ( G ) + 2 Proof. By Lemma 1, Rank (( N G ( H ) /H ) ≀ S α ) ≤ Rank( N G ( H )) + 2. Observe that Rank( G ) ≤ ℓ ( G ),as the set { g i ∈ G : g i ∈ G i − G i − , i = 1 , . . . , ℓ } (with G i as the above chain of proper subgroups)generates G . Moreover, it is clear that ℓ ( K ) ≤ ℓ ( G ) for every subgroup K ≤ G , so the result followsby letting K = N G ( H ). Theorem 4. Let G be a finite group of size n , r := r ( G ) , and A a finite set of size q ≥ . Let r i bethe number of conjugacy classes of subgroups of G of index i . Let p , . . . , p s be the prime divisorsof | G | and let r P = P si =1 r i . Then: Rank(ICA( G ; A )) ≤ ( ( r − r P − ℓ ( G ) + 2 r − r − if q = 2 , ( r − r P − ℓ ( G ) + 2 r if q ≥ . Proof. Let H , H , . . . , H r be the list of different subgroups of G with H r = G . By Theorem 1 andLemmas 1, 6, 7, Rank(ICA( G ; A )) ≤ r − X i =1 Rank (( N G ( H i ) /H i ) ≀ S α i ) + Rank( S q ) ≤ X [ G : H i ]= p k X =[ G : H i ] = p k ( ℓ ( G ) + 2) + 2= 2 r P + ( r − r P − ℓ ( G ) + 2) + 2= ( r − r P − ℓ ( G ) + 2 r. q = 2, we may improve this bound as follows:Rank(ICA( G ; A )) ≤ X [ G : H i ]=2 X [ G : H i ]= p k =2 X < [ G : H i ] = p k ( ℓ ( G ) + 2) + 1= r + 2( r P − r ) + ( r − r P − ℓ ( G ) + 2) + 1= ( r − r P − ℓ ( G ) + 2 r − r − . If G is a subgroup of S n , we may find a good upper bound for Rank(ICA( G ; A )) in terms of n by using a theorem of McIver and Neumann. Proposition 1. Suppose that G ≤ S n , for some n > . Let r := r ( G ) . Then Rank(ICA( G ; A )) ≤ ( ( r − (cid:4) n (cid:5) + 2 r − r − if q = 2 , ( r − (cid:4) n (cid:5) + 2 r if q ≥ . Proof. By [12], for every n > K ≤ S n , Rank( K ) ≤ ⌊ n ⌋ . The rest of the proof isanalogous to the previous one. Example 3. Consider the symmetric group S . In this case it is known that r = r ( S ) = 11 and r = 1 (as A is its only subgroup of index 2). Therefore,Rank(ICA( S ; A )) ≤ ( (11 − + 2 · − − q = 2 , (11 − + 2 · 11 = 42 if q ≥ . For sake of comparison, the group ICA( S ; { , } ) has order 2 . Proposition 2. Let G be a finite group and A a finite set of size q ≥ . Then Rank(ICA( G ; A ) ≥ ( r ( G ) − r ( G ) if q = 2 ,r ( G ) otherwise . . Proof. Let [ H ] , [ H ] , . . . , [ H r ] be the conjugacy clases of subgroups of G , with r = r ( G ). As longas α i > 1, the factor ( N G ( H i ) /H i ) ≀ S α i , in the decomposition of ICA( G ; A ), has a proper normalsubgroup ( N G ( H i ) /H i ) ≀ A α i (where A α i is the alternating group of degree α i ). We know that α i = 1if and only if [ G : H ] = 2 and q = 2 (Lemma 2). Hence, for q ≥ 3, we haveRank(ICA( G ; A )) ≥ Rank (cid:18) Q ri =1 (( N G ( H i ) /H i ) ≀ S α i ) Q ri =1 (( N G ( H i ) /H i ) ≀ A α i ) (cid:19) = Rank r Y i =1 Z ! = r. Assume now that q = 2, and let [ H ] , . . . , [ H r ] be the conjugacy classes of subgroups of indextwo, with r = r ( G ). Now, Rank(ICA( G ; A )) is at leastRank (cid:18) Q ri =1 (( N G ( H i ) /H i ) ≀ S α i ) Q ri =1 (( N G ( H i ) /H i ) ≀ A α i ) (cid:19) = Rank r Y i = r +1 Z ! = r − r , and the result follows. 9he previous result could be refined for special classes of finite groups. In [5] this has been donefor cyclic groups, and we do it next for dihedral groups. Proposition 3. Let n ≥ and A a finite set of size q ≥ . Rank(ICA( D n ; A )) ≥ d − (2 n ) + 2d + (2 n ) if n is odd and q ≥ , d − (2 n ) + 2d + (2 n ) − if n is odd and q = 2 , d − (2 n ) + 2d + (2 n ) + 4d + ( n ) if n is even and q ≥ , d − (2 n ) + 2d + (2 n ) + 2d + ( n ) − if n is even and q = 2 ,Proof. We shall use the decomposition of ICA( D n ; A ) given in the proof of Theorem 2. For each m i | n even greater than 2, the corresponding α i is greater than 1 by Lemma 2. The group D m i ≀ S α i has a normal subgroup N ∼ = ( Z m i / ) α i such that ( D m i ≀ S α i ) /N ∼ = Z ≀ S α i . Now, Z ≀ S α i has anormal subgroup U = (( a , . . . , a α i ); id) : α i X j =1 a j = 0 mod (2) such that ( Z ≀ S α i ) /U ∼ = Z × S α i . Finally, a copy of the alternating group A α i is a normal subgroupof Z × S α i with quotient group Z × Z . This implies that D m i ≀ S α i has a normal subgroup withquotient group isomorphic to Z × Z .Suppose that n is odd and q ≥ 3. Then ICA( D n ; A ) has a normal subgroup with quotientgroup isomorphic to Y m i | n odd Z × Y 3. Then ICA( D n ; A ) has a normal subgroup with quotientgroup isomorphic to Y m i | n odd Z × Y Let G be a group that is not finitely generated. Suppose that CA( G ; A ) has a finitegenerating set H = { τ , . . . , τ k } . Let S i be a memory set for each τ i . Then G = h∪ ki =1 S i i , so let τ ∈ CA( G ; A ) be such that its minimal memory set is not contained in h∪ ki =1 S i i . As memory set forthe composition τ i ◦ τ j is S i S j = { s i s j : s i ∈ S i , s j ∈ S j } , τ cannot be in the monoid generated by H , contradicting that H is a generating set for CA( G ; A ). This shows that CA( G ; A ) is not finitelygenerated whenever G is not finitely generated. 10he next result, which holds for an arbitrary group G , will be our main tool. Lemma 8. Let G be a group and A a set. For every normal subgroup N of G , Rank(CA( G/N ; A )) ≤ Rank(CA( G ; A )) . Proof. By [8, Proposition 1.6.2], there is a monoid epimorphism Φ : CA( G ; A ) → CA( G/N ; A ).Hence, the image under Φ of a generating set for CA( G ; A ) of minimal size is a generating set forCA( G/N ; A ) (not necesarily of minimal size). Theorem 5. Let G be an infinite abelian group and A a finite set of size q ≥ . Then, the monoid CA( G ; A ) is not finitely generated.Proof. If G is not finitely generated, then Remark 1 shows that CA( G ; A ) is not finitely generated,so assume that G is finitely generated. By the Fundamental Theorem of Finitely Generated AbelianGroups, G is isomorphic to Z s ⊕ Z p ⊕ Z p ⊕ · · · ⊕ Z p t , where s ≥ G is infinite), and p , . . . , p t are powers of primes. Then, for every k ≥ 1, wemay find a subgroup N ∼ = h k i ⊕ Z s − ⊕ Z p ⊕ Z p ⊕ · · · ⊕ Z p t such that G/N ∼ = Z k . By Lemma 8 and Proposition 2,Rank(CA( G ; A )) ≥ Rank(CA( Z k ; A )) ≥ Rank(ICA( Z k ; A )) ≥ r ( Z k ) − k. As the above holds for every k ≥ 1, then CA( G ; A ) is not finitely generated.The abelianization of any group G is the quotient G/ [ G, G ], where [ G, G ] is its commutatorsubgroup , i.e. the normal subgroup of G generated by all commutators [ g, h ] := ghg − h − , g, h ∈ G .The abelianization of G is in fact the largest abelian quotient of G . Corollary 2. Let G be a group with an infinite abelianization and A a finite set of size q ≥ .Then, the monoid CA( G ; A ) is not finitely generated.Proof. Let G ′ = G/ [ G, G ] be the abelianization of G . By Lemma 8, we have Rank(CA( G ′ ; A )) ≤ Rank(CA( G ; A )). But CA( G ′ ; A ) is not finitely generated by the previous theorem, so the resultfollows. Corollary 3. Let F S be a free group on a set S and A a finite set of size q ≥ . Then, the monoid CA( F S ; A ) is not finitely generated.Proof. As F S has an infinite abelianization, which is the free abelian group on S , the result followsby the previous corollary.The infinite dihedral group D ∞ = h x, y | x = y = id i has finite abelianization Z ⊕ Z .However, we can still show that CA( D ∞ , A ) is not finitely generated. Proposition 4. Let A be a finite set of size q ≥ . Then, CA( D ∞ ; A ) is not finitely generated.Proof. For every n ≥ 1, define H n := h ( xy ) n i ≤ D ∞ , which is a normal subgroup of D ∞ withquotient group D ∞ /H n ∼ = D n . By Proposition 2, As r ( D n ) = 1, for every n ≥ D ∞ ; A )) ≥ Rank(CA( D n ; A )) ≥ r ( D n ) − , We know that r ( D n ) ≥ d(2 n ), so, taking n = 2 k − , k ≥ 1, we see that Rank(CA( D ∞ ; A )) ≥ d (2 k ) − ≥ k , for any k ≥ 1. 11 uestion 1. Is there an infinite group G such that CA( G ; A ) is finitely generated?The techniques of this section seem ineffective to answer this for infinite groups with few properquotients, such as the infinite symmetric group. The second author thanks the National Council of Science and Technology (CONACYT) of theGovernment of Mexico for the National Scholarship (No. 423151) which allowed him to do part ofthe research reported in this article. References [1] Ara´ujo, J., Schneider, C.: The rank of the endomorphism monoid of a uniform partition. Semi-group Forum , 498–510 (2009).[2] Boyle, M., Lind, D., Rudolph, D.: The Automorphism Group of a Shift of Finite Type. Trans.Amer. Math. Soc. , no. 1, (1988).[3] Cameron, P.J.: Permutation Groups. London Mathematical Society Student Texts , Cam-bridge University Press, 1999.[4] Cameron, P.J., Solomon, R., Turull, A.: Chains of subgroups in symmetric groups. J. Algebra (2), 340–352 (1989).[5] Castillo-Ramirez, A., Gadouleau, M.: Ranks of finite semigroups of one-dimensional cellularautomata. Semigroup Forum , no. 2, 347–362 (2016).[6] Castillo-Ramirez, A., Gadouleau, M.: On Finite Monoids of Cellular Automata. In: Cook, M.,Neary, T. (eds.) Cellular Automata and Discrete Complex Systems. LNCS ∼ kconrad/blurbs/grouptheory/dihedral2.pdf.[10] Gomes, G.M.S., Howie, J.M.: On the ranks of certain finite semigroups of transformations.Math. Proc. Camb. Phil. Soc. , 395–403 (1987).[11] Gray, R.D.: The minimal number of generators of a finite semigroup. Semigroup Forum ,135–154 (2014).[12] McIver, A., Neumann P.: Enumerating finite groups, Quart.J. Math. Oxford38