Rigid commutators and a normalizer chain

The novel notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on 2^n letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler's partition theorem.

representing the partial sums of the famous sequence {b j } of the number of partitions of the integer j into at least two distinct parts, already studied by Euler [Eul48], and a sequence of group-theoretical invariants. Our sequence arises in connection with a problem in algebraic cryptography, namely the study of the conjugacy classes of affine elementary abelian regular subgroups of the symmetric group on 2 n letters [CDVS06,CCS17,ACGS19]. This is relevant in the cryptanalysis of block ciphers, since it may trigger a variation of the well-known differential attack [BS91]: a statistical attack which allows to recover information on the secret unknown key by detecting a bias in the distribution of the differences on a given set of ciphertexts when the corresponding plaintext difference is known. In particular, if F n 2 serves as the message space of a block cipher (see e.g. [DR13]) which has been proven secure with respect to differential cryptanalysis [NK95] and if T represents the translation group on F n 2 , any conjugate of T can be potentially used to define new alternative operations on F n 2 for a successful differential attack [CBS19]. In [ACGS20], on the basis of the aforementioned interest, the authors studied a chain of normalizers, which begins with the normalizer N 0 n of T in a suitable Sylow 2-subgroup Σ n of Sym(2 n ) and whose i-th term N i n is defined as the normalizer in Σ n of the previous one. After providing some experimental as well as theoretical evidence, the authors conjectured [ACGS20, Conjecture 1] the number log 2 N i n : N i−1 n to be independent of n for 1 ≤ i ≤ n − 2, and to be equal to the (i + 2)-th term of the sequence of the partial sums of the sequence {b j } 1 previously mentioned [OEI, https://oeis.org/A317910].
In this paper we completely settle this conjecture. The first attempts to solve this problem were based on theoretical techniques which clashed with their own growing computational complexity. For this reason, we develop here a novel framework to approach the problem from a different point of view. In this new approach, indeed, we take into account both the imprimitivity and the nilpotence of the Sylow 2-subgroup Σ n to represent its elements in terms of a special family of left-normed commutators, that we call rigid commutators, in a fixed set of generators. Any such commutator [X] can be identified with a subset X of {1, . . . , n}. The subgroups of Σ n that can be generated by rigid commutators are called here saturated subgroups. A careful inspection led us to prove that the normalizers N i n are saturated subgroups. In particular, a set of generators of N i n can be obtained from a set of generators of N i−1 n by adding the rigid commutators of the form [X] for all X such that the elements of the complementary set of X in {1, . . . , k}, where k = max X ≤ n, yield a partition of i + 2 − n + k into at least two distinct parts. This is the key to prove the conjecture.
The advantage of adopting rigid commutators is twofold. In the first place, they prove to be handy in calculations with the use of the rigid commutator machinery, a dedicated set of rules which we develop in this paper. Secondly, rigid commutators can be seen as factors in a unique factorization formula for the elements of any given saturated subgroup. This representation is crucial in showing that the normalizers N i n are saturated. By means of this result and of the machinery, we derive an algorithm which efficiently computes the normalizer chain.
The paper is organized as follows: in Section 1 some basic facts on the Sylow 2-subgroup Σ n of Sym(2 n ) are recalled. Section 2 is totally devoted to the introduction and the study of rigid commutators and to the construction of the rigid commutator machinery. In Section 3 the rigid commutator machinery is used to prove the conjecture on the normalizer chain previously mentioned [ACGS20, Conjecture 1]. In Section 4 it is shown that each term of the normalizer chain is a saturated group and an efficient procedure to determine the rigid generators of the normalizers is derived. An explicit construction of the normalizer chain in a specific case is provided in Section 5, and some open problems arising from computational evidence are discussed. Finally, some hints for future investigations are presented in Section 6.
1. The Sylow 2-subgroup of Sym(2 n ) Let n be a non-negative integer. We start recalling some well-known facts about the Sylow 2-subgroup Σ n of the symmetric group on 2 n letters. 1 The sequence b j + 1 appears in several others areas of mathematics, from number theory to commutative algebra [ES14]. In particular, it was already known to Euler that b j + 1 corresponds to the number of partitions of j into odd parts (see [Eul48,Chapter 16] and [And07,§3]). Several proofs of this Euler's partition theorem have been offered ever since [SF82,And94,KY99], and several important refinements have been obtained [SF82,Fin88,Bes94,BME97,Str16] Let us consider the set of binary words of length n, where T 0 contains only the empty word. The infinite rooted binary tree T is defined as the graph whose vertices are j≥0 T j and where two vertices, say w 1 . . . w n and v 1 . . . v m , are connected by an edge if |m − n| = 1 and w i = v i for 1 ≤ i ≤ min(m, n). The empty word is the root of the tree and it is connected with both the two words of length 1. We can define a sequence {s i } i≥1 of automorphisms of this tree. Each s i necessarily fixes the root, which is the only vertex of degree 2. The automorphism s 1 changes the value w 1 of the first letter of every non-empty word intow 1 def = (w 1 + 1) mod 2 and leaves the other letters unchanged. If i ≥ 2, we define (1.1) In general, s i leaves a word unchanged unless the word has length at least i and the letters preceding the i-th one are all zero, in which case the i-th letter is increased by 1 modulo 2. If i ≤ n and the word w 1 . . . w n ∈ T n is identified with the integer 1 + n i=1 2 n−i w i ∈ {1, . . . , 2 n }, then s i acts on T n as the the permutation whose cyclic decomposition is which has order 2. In particular, the group s 1 , . . . , s n acts faithfully on the set T n , whose cardinality is 2 n , as a Sylow 2-subgroup Σ n of the symmetric group Sym(2 n ) (see also Fig. 1).
It is also well known that Σ n = s n ≀ Σ n−1 = s n ≀ · · · ≀ s 1 ∼ = ≀ n i=1 C 2 is the iterated wreath product of n copies of the cyclic group C 2 of order 2.
The support of a permutation is the set of the letters which are moved by the permutation. We say that two permutations σ and τ are disjoint if they have disjoint supports; two disjoint permutations always commute. The is generated by disjoint conjugates of s i , hence S i is an elementary abelian 2-group which is normalized by S j if j ≤ i. Moreover, Σ n = S 1 ⋉ · · · ⋉ S n ∼ = Σ n−1 ⋉ S n .

Rigid commutators
The commutator of two elements h and k in a group G is defined as The left-normed commutator of the m elements g 1 , . . . , g m ∈ G is the usual commutator if m = 2 and is recursively defined by It is well known that the commutator subgroup G ′ of a finitely generated nilpotent group G can be generated by left-normed commutators involving only generators of G [Hup67, III.1.11]. From now on, we will focus on left-normed commutators in s 1 , . . . , s n . For the sake of simplicity, we write [i 1 , . . . , i k ] to denote the left-normed commutator [s i1 , . . . , s i k ], when k ≥ 2, and we also write [i] to denote the element s i . Definition 1. A left-normed commutator [i 1 , . . . , i k ] is called rigid, based at i 1 and hanging from i k , if i 1 > i 2 > · · · > i k . Given a subset X = {i 1 , . . . , i k } ⊆ {1, . . . , n} such that i 1 > i 2 > · · · > i k , the rigid commutator indexed by X, denoted by [X], is the left-normed commutator [i 1 , . . . , i k ]. We set [X] def = 1 when X = ∅. The set of all the rigid commutators of Σ n is denoted by R and we let At the end of this section we prove that every permutation in the Sylow 2subgroup Σ n can be expressed, in a unique way, as a product of the objects previously defined. To this purpose, we develop below a set of rules to perform computations with (rigid) commutators.    Fact 3. If k ≥ 2 and i j = i j+1 for some 1 ≤ j ≤ k − 1, then [i 1 , . . . , i k ] = 1.
The following result is crucial since it allows us to rewrite every commutator as a rigid commutator.
Proof. Let us prove each claim separately.
(1) The claim c 2 = 1 depends on the fact that c ∈ S i , where the index i is defined as i def = max {i 1 , . . . , i k , j 1 , . . . , j l }.
(3) Assume that l ≥ 2 and j l < i k . In this case The permutations s j l [j 1 , . . . , j l−1 ]s j l and [i 1 , . . . , i k ] [j1,...,j l−1 ] are disjoint: the first one has support contained in 2 n−j l + 1, . . . , 2 n−j l +1 and the support of the second one is contained in which proves the claim.
(4) The claim follows by a repeated applications of items (3) and (1).
(5) For every x, y ∈ G def = s n , . . . , s i l +1 the permutations x and y sj l are disjoint and so they commute. In particular, if as required.
It is worth noticing here that rigid commutators are the images of P. Hall's basic commutators [Hal34] under the presentation of the group Σ n as a factor of the n-generated free group, once the order of the generators is reversed.
2.2. Saturated subgroups. In this section we give a representation of the elements of Σ n in terms of rigid commutators. Proof. Let 1 ≤ i ≤ n. To prove the claim, we look at S i as a 2 i−1 -dimensional vector space over F 2 . Proceeding by backward induction on j, for i ≥ j ≥ 1, we show that the set of all the rigid commutators based at i and hanging from h, for some h ≥ j, is linearly independent. When j = i there is nothing to prove. Assume where the exponents are in F 2 . We aim at proving that all the exponents are 0.
Note that if the permutation on the right-hand side of Eq. (2.2) is non-trivial, then it moves some x with x > 2 n−j , which is fixed by the one on the left-hand side.
Hence the permutations on both sides are trivial. By induction, the exponents in the left-hand side of Eq. (2.2) are all 0. Now, the commutator map Again, by the inductive hypothesis, we find e i,i1,...,it−1,j = 0 for every choice of i 1 > · · · > i t−1 . As the number of rigid commutators based at i equals the dimension of S i , the proof is complete.
We can now state our first main result as a straightforward consequence of Lemma 4. Let us call a proper order ≺ on R * any total order refining the partial order defined by [i 1 , . . . , i t ] ≺ [j 1 , . . . , j l ] if i 1 < j 1 . Here we denote by P n the power set of {1, . . . , n}.
Theorem 5. Given a proper order ≺ in R * , every element g ∈ Σ n can be uniquely represented in the form where the factors are ordered with respect to ≺ and e g : P n \ {∅} → {0, 1} is a function depending on g.
Proof. Since Σ n = S 1 ⋉ · · · ⋉ S n , the claim is a straightforward consequence of Lemma 4.
Some of the following corollaries are straightforward and their proof will be omitted.
Corollary 6. If G is a subgroup of Σ n containing k distinct rigid commutators, then |G| ≥ 2 k .
We now need a new concept which plays a key role in the remainder of this work.
Remark 1. A subgroup G ≤ Σ n is saturated if and only if it can be generated by some subset X of R: indeed G is also generated by the smallest saturated subset of G ∩ R containing X .
Corollary 8. Let G ≤ Σ n be a saturated subgroup generated by a saturated set G ⊆ R * and let ≺ be any given proper order on G. Every element g ∈ G has a unique representation where the commutators in the product are ordered with respect to ≺ and e c (g) ∈ {0, 1}. In particular |G| = 2 |G| .
Corollary 9. Let G ≤ Σ n be a saturated subgroup generated by a saturated set G ⊆ R * and let ≺ any given proper order on G. If the product c 1 · · · c k ∈ G, where c i ∈ R * , and c 1 c 2 · · · c k , then c i ∈ G for all 1 ≤ i ≤ k.
Proof. Note that since every rigid commutator belongs to some S i , the group G has the semidirect product decomposition G = (G ∩ S 1 ) ⋉ · · · ⋉ (G ∩ S n ). In particular every element of G can be written as an ordered product of elements of G. Write c 1 · · · c k = g 1 · · · g t where g i ∈ G and g 1 · · · g t . By Theorem 5, we have k = t and c i = g i ∈ G.
The next statement follows immediately from Corollary 9.
Corollary 10. Let G ≤ Σ n be a saturated subgroup. If g = g 1 · · · g n , where

Elementary abelian regular 2-groups and their chain of normalizers
A vector space T of dimension n over F 2 acts regularly over itself as a group of translations. By way of this action, T can be seen as a regular elementary abelian subgroup of Sym(2 n ), and any other regular elementary abelian subgroup of Sym(2 n ) is conjugate to T in Sym(2 n ) [Dix71]. The normalizer of T in Sym(2 n ) is the affine group AGL(T ), where T embeds as the normal subgroup of translations. For this reason, we refer to any of the conjugates of T as a translation subgroup of Sym(2 n ). Every chief series is normalized by exactly one Sylow 2-subgroup U F of AGL(T ). In [Lei88, Theorem p. 226] it is proved that every chief series F of T corresponds to a Sylow 2-subgroup Σ F of Sym(2 n ) containing T and having a chief series that intersects T in F. The correspondence F → Σ F is a bijection between the sets of the chief series of T and the set of Sylow 2-subgroups of Sym(2 n ) containing T . In [ACGS20] it is also pointed out that U F = N Σ F (T ) = Σ F ∩ AGL(T ). From now on the chief series F will be fixed, and so, without ambiguity, we will write Σ n and U n to denote respectively Σ F and U F . In [ACGS19] it is proved that U n contains, as normal subgroups, exactly two conjugates of T , namely T and T Un = T g , for some g ∈ Sym(2 n ). It is also shown that the normalizer N 1 n = N Sym(2 n ) (U n ) interchanges by conjugation these two subgroups and that N 1 n contains U n as a subgroup of index 2. In particular, N 1 n ≤ Σ n . In the following section we will extend these results on T, U n , N 1 n to the entire chain of normalizers, which is defined below.
3.1. The normalizer chain. The normalizer chain starting at T is defined as In [ACGS20] the authors proved that N Σn (N i n ) = N Sym(2 n ) (N i n ), for all i ≥ 0, computed the normalizer chain for n ≤ 11 by way of the computer algebra package GAP [GAP20], and conjectured that the index N i+1 n : N i n does not depend on n for n ≥ i + 3 [ACGS20, Conjecture 1]. In this section we prove this conjecture arguing by induction, by means of the rigid commutator machinery developed in Section 2.1. We start by defining Lemma 11. T is an elementary abelian regular subgroup of Σ n . In particular, T is a translation subgroup of Sym(2 n ).
Proof. T is a subgroup of Σ n as it is generated by elements belonging to Σ n . By item 6 of Lemma 2 it follows that [t i , t j ] = 1, so that T is abelian. Note that t 2 i = 1 as t i ∈ S i , and so T is elementary abelian of order at most 2 n . Let us now prove that T is transitive. Let 1 ≤ x ≤ 2 n be an integer represented as x = 1 + n i=1 2 n−i w i in binary form and let t = n i=1 t wi i . A direct check shows that t moves 1 to x. Since T has an orbit with 2 n elements and it has order at most 2 n , it follows that |T | = 2 n and that every point stabilizer is trivial, therefore T is a regular permutation group on {1, . . . , 2 n }.
Let us now determine the permutations in Σ n normalizing T . For 1 ≤ j < i ≤ n let us define X ij From now on we will set Proposition 12. The group U n is the normalizer of T in Σ n , i.e.
Proof. Let us set U def = T, u ij | 1 ≤ j < i ≤ n and let us prove that U = U n = N Σn (T ). By Lemma 2 we have This shows that U ≤ N Σn (T ) = U n and that U n is a saturated set. Therefore, from Corollary 8, |U | = 2 |Un| = 2 n(n+1) 2 = |U n |, which proves the claim.
We aim at proving our second main result, providing the generators of the normalizer N i n in terms of rigid commutators. The result is proved by induction on i ≥ 1. (3.2) We now prove that we can generate N 1 n by appending η n to the list U n of the rigid commutators generating U n .
Proposition 13. If n ≥ 3, then the group U n , η n is the normalizer N 1 n of U n in Σ n , i.e. N 1 n = T, u ij , η n | 1 ≤ j < i ≤ n . Moreover, N 1 n : U n = 2. Proof. By Lemma 2, and [u ij , η n ] = u n,1 if i = 2 and j = 1 1 otherwise , Thus the rigid commutator η n belongs to N Σn (U n ), hence U n , η n ≤ N Σn (U n ). Moreover U n ∩ S n = t n , u n,1 , . . . , u n,n−1 and so η n , which is based at n, is such that η n / ∈ U n . The claim now follows from |N Σn (U n ) : U n | = 2 [ACGS19, Theorem 7]. for each 1 ≤ i ≤ n and j, and Note that, if j ≤ i − 2, then |W i,j | = b j , i.e. the number of partitions of j into at least two distinct parts. Our next goal is to prove that N i n = N i n for each 0 ≤ i ≤ n − 2, where N i n is defined as in Eq. (3.1). Propositions 12 and 13 show that this is actually the case when i ∈ {0, 1}.
In order to prove the general result, we need the following reformulation of item 6 of Lemma 2 to compute commutators of rigid commutators written in punctured form. . If a = b, then c = 1. Without loss of generality, we can assume that a > b. By Lemma 2, if b / ∈ I, then c = 1. If b ∈ I, the claim follows from item 6 of Lemma 2.
In the following facts, we summarize some properties that will be useful in the proof of the conjecture. (3) |J| ≥ 2, and j∈J j ≤ i + 2 − (n − a).
Fact 5. Note that for 2 ≤ i ≤ n − 2 the set N i n ∩ (S 1 ⋉ · · · ⋉ S n−1 ) is equal to N i−1 n−1 . Indeed, at the i-th iteration, the newly generated elements of N i n , which are those in N i n \ N i−1 n , are constructed by lifting the elements of N i−1 n \ N i−2 n , i.e. by replacing a rigid commutator based at j with the rigid commutator obtained by removing its left-most element, for j ≤ n, and by adding some new rigid commutators based at n, in accordance with Eq. (3.5). Proceeding in this way it is easy to check that, disregarding all the commutators based at n in N i n , the lifted elements are exactly the elements of N i−1 n−1 . The reader is referred to Section 5 for explicit examples. Fact 6. In the proof of Proposition 13 we showed that [ N 1 } and using Fact 5, we can conclude that Similarly, Proof. The claim follows from Lemma 15, Fact 6 and Corollary 8.
We conclude this section with our main result showing that the i-th term of the normalizer chain is actually generated by the set N i n of rigid commutators defined in Eq. (3.5). We prove, indeed, that the inclusion N i n ≤ N Σn N i−1 n shown in the previous proposition is actually an equality.
Theorem 17. For i ≤ n− 2, the group N i n is the i-th term N i n of the normalizer chain.
Proof. The cases i = 0 and i = 1 has been addressed respectively in Propositions 12 and 13. We assume by induction on i ≥ 2 that N j m = N j m for all m ≤ n whenever j < i ≤ m − 2. In particular where the first equality in the last line holds since the following inclusions have trivial intersection and that S n = A × B. By Lemma 15 we have that A is a subgroup of N i n ∩ S n , for 1 ≤ j ≤ i, so that N i We denote a generic element of H by  Table 1.  Table 1. First values of the sequences a i and b i We have developed the rigid commutator machinery as a theoretical tool of investigation. It is not surprising anymore that the equality b i = |W n,i |, where W n,i is defined by Eq. (3.4), is the link with the mentioned sequence. This combinatorial identity, Eq. (1.1), Proposition 16 and Theorem 17 give at last a positive answer to Conjecture 1 in [ACGS20].
Corollary 18. For 1 ≤ i ≤ n − 2, the number log 2 N i n : N i−1 n is independent of n. It equals the (i + 2)-th term of the sequence {a j } of the partial sums of the sequence {b j } counting the number of partitions of j into at least two distinct parts.

Normalizers of saturated subgroups
In this section we will prove that the normalizer N def = N Σn (G) of a saturated subgroup G of Σ n is also saturated, provided that T ≤ N , and thus we can use our rigid commutator machinery in the computation of N . In particular, for i ≤ n − 2, the machinery could be used as an alternative tool to derive the theoretical description of N i n as in Theorem 17. Even if we do not have such a description when i > n − 2, the machinery can be anyway used to efficiently compute via GAP the complete normalizer chain. We denote below by N i the intersection N ∩ S i .
Proposition 19. If G is a saturated subgroup of Σ n , and N = N Σn (G) is its normalizer in Σ n , then Proof. Let x ∈ N and write x = x i1 . . . x i k where 1 ≤ i 1 < · · · < i k ≤ i k+1 def = n and x ij ∈ S ij , for 1 ≤ j ≤ k. In order to prove our claim we first show that [x i1 , c] ∈ G for every non-trivial rigid commutator c of G. Since G is generated by its own non-trivial rigid commutators, it will follow that x i1 ∈ N . As a consequence, also x i2 . . . x i k ∈ N . Thus, we may argue by induction on k to obtain that x ij ∈ N for all 1 ≤ j ≤ k.
Let i be such that c ∈ G ∩ S i . Suppose first that i < i 1 . If [c, x i1 ] ∈ S i1 and h ∈ t>i1 S t . By Corollary 10 we obtain that [c, x i1 ] ∈ G ∩ S i1 ≤ G. If i = i 1 , then trivially [c, x i1 ] = 1 ∈ G. The last possibility is i 1 < · · · < i m < i ≤ i m+1 for some m ≤ k. Suppose that [x i1 , c] = 1. In this case
Lemma 20. Suppose that G is a saturated subgroup of Σ n normalized by T . If x 1 , . . . , x k ∈ S j are distinct rigid commutators such that x = x 1 · · · x k ∈ N , then Proof. Let c 1 , . . . , c h ∈ R * such that G = c 1 , . . . , c h and let us write every c s and x t in punctured form: Suppose first that m s < j, so that Notice that if the commutator d s,t appears only once in the product, then, by Corollary 9, d s,t ∈ G. If C s ∩ X t = ∅ for all 1 ≤ t ≤ k, then all the non-trivial d s,t appearing in the product are distinct and hence they appear only once in the product, so that d s,t ∈ G for all 1 ≤ t ≤ k. If C s ∩ X t = ∅, then the commutator d s,t may appear more than once in the product displayed in Eq. (4.1). Let l ∈ C s ∩ X t and consider the commutator Let C = C s \{l}. We have determined a new rigid commutator c = c s,l = ∨[m s ; C] ∈ G such that |C ∩ X t | < |C s ∩ X t |, that |C| < |C s | and that d s,t = [c, x t ] appears in the expansion of [c, x]. Using the same strategy, after a finite number of steps, we obtain c = ∨[m s ; C] ∈ G such that C ∩ X t = ∅. If d s,t = [c, x t ] = [c, x t1 ] = d s,t1 , for some t 1 = t, then C ∩ X t1 = ∅, since otherwise X t = X t1 and consequently x t = x t1 with t = t 1 , contrary to the hypotheses. Thus we may proceed in the same way with d s,t1 . Since at each step the cardinality of C is strictly decreasing, after a finite number of steps we find a c ∈ G and x tr such that d s,t = d s,t1 = · · · = d s,tr appears only once in [c, x] giving d s,t ∈ G. This finally shows that d s,t ∈ G for all 1 ≤ t ≤ k.
If j = m s then x i and c s commute for all i and there is nothing to prove.
We are left with the case when m s > j. As above, we have Reasoning as we did for m s < j, we obtain that d s,t ∈ G for all 1 ≤ t ≤ k.
In all the cases we have proved that x i ∈ N for all 1 ≤ i ≤ k, which is our claim.
As an easy consequence of Proposition 19 and Lemma 20 we find the following result.
Theorem 21. The normalizer N in Σ n of a saturated subgroup of Σ n is also saturated, provided that N contains T . Remark 4. Another proof of Theorem 17 can be obtained by Theorem 21. Indeed, it is not difficult, but rather tedious, to check that for 0 ≤ i ≤ n − 2. The result then follows by Proposition 16.
From Theorems 17 and 21 and from Remark 4 we derive a straightforward corollary resulting in an algorithm whose GAP implementation is publicly available at GitHub 2 . This script allows a significant speed-up in the computation of the normalizer N of a saturated subgroup provided that N contains T . We could easily apply this script to compute our normalizer chain up to the dimension n = 22. For example, whereas the standard libraries required one month on a cluster to compute the terms of the normalizer chain in Sym(2 10 ), our implementation of the rigid commutator machinery gives the result in a few minutes, even on a standalone PC. With a similar approach, we can also use rigid commutators to compute the normal closure of a saturated subgroup. Some explicit calculations are shown below in Section 5. Let M i n be the set of all the rigid commutators belonging to N i n . From Theorem 21, the subgroups N i n are saturated, hence N i n = M i n for all i ≥ 1. Corollary 22. The set M i n is the largest subset of R that normalizes M i−1 n , i.e.
The construction of the terms of the normalizer chain is then reduced to the determination of the sets M i n , a task which turns out to be way faster than computing the terms of the normalizer chains as subgroups of Σ n via the Normalizer command provided by GAP.

A computational supplement
In this section we show an explicit construction of the first four groups in the normalizer chain when n = 6, i.e. in Sym(64). Let us start with determining the generators of T in terms of rigid commutators: We have that T = t 1 , t 2 , . . . , t 6 and, from Proposition 12, its normalizer in Σ n is N 0 consequently |N 0 6 | = 2 21 . Now, in accordance with Eq. (3.5) and Theorem 17, the normalizer N 1 6 is generated by the rigid commutators previously listed and by η 6 , the only element of W 6,3 (see Eq. (3.4)). The commutator η 6 is the punctured rigid commutator based at 6 and missing the integers 1 and 2, i.e. where 1 and 2 indeed represent the sole partition of 3 into at least two distinct parts. From this, log 2 N 1 6 : N 0 6 = 1 = a 3 . Again from Eq. (3.5) and Theorem 17, the normalizer N 2 6 is generated, along with the elements already mentioned, by the rigid commutators in W 5,3 and W 6,4 , i.e. The commutator of Eq. (5.2), which belongs to W 5,3 , is the punctured rigid commutator based at 5 and missing the integers 1 and 2. complete the set of rigid generators of N 4 6 , and log 2 N 4 6 : N 3 6 = 7 = a 6 .
Using Corollary 22, we can find a saturated set of rigid generators for all the elements of the chain. Notice that for i > 5, the sequence log 2 N i 6 : N i−1 6 does not fit the pattern of the sequence {a j }. Although we do not have a general formula to calculate the values of the relative indices between two consecutive terms in the normalizer chain, they can be explicitly determined by the algorithm in GitHub. Computational results are summarized in Table 2, where we list all the relative indices of the normalizer chain. In the second column, the logarithms of the sizes of the intersections of each term with each of the subgroups S 6 , . . . , S 1 are displayed.

Problems for future research
We conclude this work by highlighting some further properties and structures of the set R of rigid commutators and providing some hints for future research.
Algebras of rigid commutators. The operation of commutation in R is commutative and [∅] represents the zero element. Moreover, for every x, y ∈ R the following identity is satisfied Let F be any field of characteristic 2 and let r be the vector space over F having the set R * of the non-trivial rigid commutators as a basis. The space r is endowed with a natural structure of an algebra. The product x ⋆ y of two rigid commutators x, y ∈ R is defined as otherwise. This operation is then extended to the whole r by bilinearity and turns r into a Jordan algebra, since it is commutative and x ⋆ x = 0 for all x ∈ r. Moreover, if H is a saturated subset of R * , then, on the one hand the group H = H is a saturated subgroup of Σ n and, on the other hand, the F-linear span h of H is a subalgebra of r. The property [R, H] ⊆ H ∪ {[∅]} is a necessary and sufficient condition for H to be a normal subgroup of Σ n and for h to be an ideal of r. We point out that the fact that R is closed under commutation is crucial to check the previous statement. If c is the nilpotency class of Σ n , then the product of c + 1 elements of r is always zero, so that r is nilpotent. The study of the properties and the representations of this algebra seems to be a problem of independent interest, in connection with the study of the saturated subgroups of Σ n .