A Chain of Normalizers in the Sylow 2 -subgroups of the symmetric group on 2 n letters
Riccardo Aragona, Roberto Civino, Norberto Gavioli, Carlo Maria Scoppola
aa r X i v : . [ m a t h . G R ] A ug A CHAIN OF NORMALIZERS IN THE SYLOW -SUBGROUPS OF THESYMMETRIC GROUP ON n LETTERS
RICCARDO ARAGONA, ROBERTO CIVINO, NORBERTO GAVIOLI, AND CARLO MARIA SCOPPOLA
Abstract.
On the basis of an initial interest in symmetric cryptography, in the present workwe study a chain of subgroups. Starting from a Sylow 2-subgroup of AGL(2 , n ), each termof the chain is defined as the normalizer of the previous one in the symmetric group on 2 n letters. Partial results and computational experiments lead us to conjecture that, for largevalues of n , the index of a normalizer in the consecutive one does not depend on n . Indeed,there is a strong evidence that the sequence of the logarithms of such indices is the one of thepartial sums of the numbers of partitions into at least two distinct parts. Introduction
Let n be a non-negative integer and let Sym(2 n ) denote the symmetric group on 2 n letters.The study of the conjugacy class in Sym(2 n ) of the elementary abelian regular 2-subgroups hasrecently drawn attention for its application to block cipher cryptanalysis, and in particular todifferential cryptanalysis [BS91]. The reader which is familiar with symmetric cryptography willnot find hard to realize that the key-addition layer of a block cipher (see e.g. [DR13, BKL + F n of the block cipher,which can be used to perform algebraic and statistical attacks. Indeed, although the encryptionfunctions, in order to be secure, are designed to be far from being linear with respect to theclassical bitwise addition modulo 2, it is possible to attack the encryption scheme by means ofa variation of the classical differential attack, where instead a newly designed operation is used[CBS18]. Such operation is defined starting from a conjugate of the translation group T on themessage space.A study of regular subgroups of the affine group is carried out in [CDVS06, CR09] by meansof radical algebras. We point out that there is an interesting connection between our study ofthe position of a regular subgroup in the symmetric group, in terms of the chain of normalizersdefined below, and the rather recent theory of braces , introduced in [Rum07], since the abovementioned new operation can be used to construct a brace on T . Indeed, when + and ◦ respec-tively denote the (additive) laws induced by T and by one of its affine conjugates, the structure( T, + , ◦ ) is a two-sided brace and ( T, + , · ) is a radical ring, where a · b is defined as a + b + a ◦ b for each a, b ∈ T . For an extensive survey and detailed references on braces see e.g. [Ced18]. Mathematics Subject Classification.
Key words and phrases.
Symmetric group on 2 n elements; Elementary abelian regular subgroups; Sylow2-subgroups; Normalizers.All the authors are members of INdAM-GNSAGA (Italy). R. Civino is partially funded by the Centre ofexcellence ExEMERGE at University of L’Aquila. Part of this work has been carried out during the cycle ofseminars “Gruppi al Centro” organized at INdAM in Rome. n ) 2 In a recent paper [ACGS19], we considered the elements of the conjugacy class T Sym(2 n ) whichare subgroups of the affine group AGL( T ). We showed that, if T g ∩ T has index 4 in T , thenthere exists a Sylow 2-subgroup U <
AGL( T ) containing both T g and T as normal subgroups.The normalizer N of U in Sym(2 n ) contains U as a subgroup of index 2 and interchanges T and T g by conjugation. The 2-group N is therefore contained in a Sylow 2-subgroup Σ of Sym(2 n ).Motivated by a computational evidence, we prove here that this is the general behavior. Wedefine a chain starting from U and where the k -th term N k is the normalizer in Sym(2 n ) of theprevious N k − . We show that N k is actually the normalizer of N k − in Σ, and thus the N k s forma sequence of 2-groups ending at Σ. Philip Hall, indeed, proved that Σ is self-normalizing (seee.g. [CF64]). Using the software package GAP [GAP20], we computed the normalizer chain for n ≤
11. We experimentally noticed that the sequence defined by c k = log | N k : N k − | doesnot depend on n if k ≤ n − { c k } k ≥ represents the sequence of partialsum of the sequence { b k +2 } k ≥ , where b k counts the number of partitions of k into at least twodistinct parts, a well-known sequence of integers [OEI, https://oeis.org/A317910 ], also appear-ing in commutative algebra problems [ES14]. For larger values of n , the computational problembecomes intractable using the standard libraries, and so its investigation requires a theoreticalapproach. For small values of k , by way of an elementary but increasingly cumbersome analysis,we show that the previous claim is true. In the general case, the claim remains an open problem.We believe that more sophisticated combinatorial and group theoretical tools could prove that,for k ≤ n −
2, the integers c k do not depend on n and are related to the sequence b k as previouslymentioned.The paper is organized as follows: in Sec. 2 we introduce the notation and provide somepreliminary results. The normalizer chain is defined in Sec. 3, which contains the main con-siderations that led us to formulate Conjecture 1. Some theoretical evidence in support of ourconjecture, i.e. Theorem 4.7, is proved in Sec. 4, where we also discuss some open problems. Toconclude, Sec. 5 is devoted to the computational aspects and contains the GAP code used forour computations. 2.
Notation and preliminaries
In this section, we recall some well known facts and a preliminary result on the imprimitivityaction of subgroups of the symmetric group on a finite set.
Definition 2.1.
Let Ω = ∅ and let G ≤ Sym(Ω) be a transitive permutation group. Animprimitivity system B for G is a G -invariant partition of Ω. The group G is primitive if G has only the trivial partitions { Ω } and the set of the singletons of Ω as imprimitivity systems.Otherwise, G is said to be imprimitive. Definition 2.2.
Let G act imprimitively on the set Ω. An imprimitivity chain B ≻ · · · ≻ B t ofdepht t is a sequence of imprimitivity systems for G acting on Ω, where B and B t are the trivialpartitions. We also require that for each B ∈ B m +1 there exists B ′ ∈ B m such that B ⊂ B ′ for0 ≤ m ≤ t − B ≻ · · · ≻ B t can be represented by its imprimitivity tree which is the rooted tree ( V, E ), where • the set of vertices V is S tm =0 B m , more precisely a vertex is a subset of Ω belonging tosome partition B i and the root vertex is Ω; • two vertices X and Y in V are connected by an edge e ∈ E if and only if there exists m such that X ∈ B m , Y ∈ B m +1 and Y ⊂ X . CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 3 In the remainder of this work, we will consider the special case of a subgroup G of the sym-metric group Sym( X n ), where X n def = { , . . . , n } .For 0 ≤ m ≤ n and 0 ≤ k ≤ m −
1, the following notation is used: • B nm,k def = { k n − m + 1 , . . . , ( k + 1)2 n − m } , and in particular X n = B n , ; • B nm def = (cid:8) B nm, , . . . , B nm, m − (cid:9) ; • for 1 ≤ i ≤ n s i def = i − Y j =1 ( j, j + 2 i − ); • t ni def = ( s i if i = nt n − i · ( t n − i ) s n if 1 ≤ i < n. The symmetric group Sym(2 n ) acts on the set of partitions of X n and, with respect to thisaction, we define the subgroupΣ n def = n \ m =1 Stab
Sym(2 n ) ( B nm ) = h s , . . . , s n i ∼ = ≀ ni =1 C , which is the n -th iterated wreath product of copies of the cyclic group C of order 2, i.e. a Sylow2-subgroup of Sym(2 n ). Notice that B n ≻ · · · ≻ B nn is an imprimitivity chain C n of maximaldepth for Σ n and that Σ n is the stabilizer of C n in Sym(2 n ).Let T n, def = { } and, for 1 ≤ i ≤ n , let us define T n,i def = h t n , . . . , t ni i . Clearly T n,i ≤ T n,i +1 ,for 0 ≤ i ≤ n −
1. The group T n def = T n,n is a regular elementary abelian subgroup of Sym(2 n )of order 2 n contained in Σ n , whose normalizer in Sym(2 n ) is AGL( T n ), the affine general lineargroup. We also define U n def = AGL( T n ) ∩ Σ n = N Σ n ( T n ) . The group T n is a uniserial module for U n whose maximal flag F n is defined as { } = T n, < · · · < T n,n = T n . Given a subgroup H ≤ Σ n − , we define the diagonal embedding of H into Σ n as∆ n ( H ) def = { ( x, x s n ) | x ∈ H } . Remark . It was already known to Dixon [Dix71] that the set of elementary abelian regularsubgroups of Sym(2 n ) form a unique conjugacy class. Moreover, a transitive abelian subgroupof Sym(2 n ) is regular and so is self-celtralizing. In particular, ( T n ) g is self-centralizing in Σ n ,for every g ∈ Sym(2 n ). Lemma 2.3.
Up to conjugation by elements of Σ n , the group T n is the unique elementary abelianregular subgroup of Sym(2 n ) having C n as imprimitivity chain.Proof. First, recall that Σ n stabilizes C n for every n . We argue by induction on n , the resultbeing trivial when n = 1. Let T be an elementary abelian regular subgroup of Sym(2 n ) having C n as imprimitivity chain and let M be the stabilizer in T of { , . . . , n − } = B n , ∈ B n . Inparticular, M stabilizes also B n , = ( B n , ) s n = { n − + 1 , . . . , n } . The group M acts on B n , as an elementary abelian regular subgroup M of S n − having C n − as imprimitivity chain. Byinduction, M = ( T n − ) h for some h ∈ Σ n − . Similarly, the group M acts faithfully on B n , as an elementary abelian regular subgroup M of ( S n − ) s n having ( C n − ) s n as imprimitivity CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 4 chain, and thus we find by induction M = (( T n − ) h ) s n for some h ∈ Σ n − . Finally, we havethat M = (cid:8) ( m h , m h s n ) | m ∈ T n − (cid:9) = ∆ n ( T n − ) ( h ,h sn ) . If t ∈ T \ M then t interchanges B n , and B n , and centralizes M . Let us write t in the form t = ( a, b s n ) s n , where a, b ∈ Σ n − and ( m h , m h s n ) = ( m h , m h s n ) t = ( m h b , ( m h a ) s n ). Notethat • t = ( a, b s n ) s n ( a, b s n ) s n = ( ab, ( ba ) s n ), and so a = b − ; • m h = m h b and m h s n = m h as n for all m ∈ T n − , from which we derive h ah − , h bh − ∈ C Σ n − ( T n − ) = T n − , i.e. a = h − uh = u h h − h and b = a − = h − h u h = u h h − h for some u ∈ T n − .Then we have t = ( a, b s n ) s n = ( u h h − h , u h s n ( h − h ) s n ) s n ≡ ( h − h , ( h − h ) s n ) s n = s ( h ,h sn ) n mod M. Since T n = ∆ n ( T n − ) ⋊ h s n i , then T = T ( h ,h sn ) n , as required. (cid:3) Remark . Notice that the chain C n is a maximal imprimitivity chain for T n , even though it isnot the only one. It is known that every maximal imprimitivity chain for T n determines and isdetermined by a maximal flag { } = T n, < · · · < T n,n = T n . Indeed, the partition B i is the set ofthe orbits of T n,n − i , and conversely T n,n − i is the pointwise stabilizer of the action of T n over B i .Any Sylow 2-subgroup U of AGL( T n ) is the stabilizer by conjugation of a maximal flag of T n , andtherefore it stabilizes also the associated imprimitivity chain. In particular, the stabilizer of C n inAGL( T n ) is U n = Σ n ∩ AGL( T n ). More generally, any maximal flag F of T n determines a Sylow2-subgroup U F of AGL( T n ) and a Sylow 2-subgroup Σ F [Lei88, Theorem p. 226] of Sym(2 n )such that U F = Σ F ∩ AGL( T n ). The maps F 7→ U F and F 7→ Σ F are injective. Consequently,for every Sylow 2-subgroup U of AGL( T n ) there exists a unique Sylow 2-subgroup Σ of Sym(2 n )such that U = Σ ∩ AGL( T n ). In particular, the intersection AGL( T n ) ∩ Σ n = N Σ n ( T n ) = U n isa Sylow 2-subgroup of AGL( T n ).3. Experimental evidence on a normalizer chain
Let us start by defining the normalizer chain of T n . Definition 3.1.
Using the same notation of the previous section, the normalizer chain of T n isdefined as the sequence { N kn } k ≥ , where N n def = U n = N Σ n ( T n ) , N n def = N Sym(2 n ) ( U n ) , and recursively, for k > N kn def = N Sym(2 n ) ( N k − n ) . Considering Σ n in place of Sym(2 n ) the resulting chain is the same, as proven in the nexttheorem. Theorem 3.2.
For every k ≥ , we have N kn = N Σ n ( N k − n ) . In particular, N kn is a -group.Proof. Suppose that B is a system of imprimitivity for N k − n . For each x ∈ N kn , the partition B x is a system of imprimitivity for ( N k − n ) x and so for N k − n , since ( N k − n ) x = N k − n . Thus, fora given x ∈ N kn and an imprimitivity chain C for N k − n , the set C x is also an imprimitivity chainfor N k − n and a fortiori for U n . Since, by Remark 2, the imprimitivity chain C n is the unique CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 5 maximal one for U n = N n , we have C xn = C n . Hence C n is stabilized by N kn for every k , yielding N kn ≤ Σ n . (cid:3) A direct consequence of the previous theorem is that there exists d ( n ) ∈ N such that N kn = N d ( n ) n = Σ n for every k ≥ d ( n ). We can interpret d ( n )+1 as an upper bound for the defect δ ( n ) of T n in Σ n , i.e.the length of the shortest subnormal series from T n to Σ n . Recalling that Σ n is self-normalizingin Sym(2 n ) (see [CF64]), as already pointed out in the introduction, the fact previously statedrepresents a further argument showing that every Sylow 2-subgroup of AGL( T n ) is contained inexactly one Sylow 2-subgroup of Sym(2 n ).We already recalled in Remark 2 that N n = U n normalizes a maximal flag F of T n . Belowwe study the action by conjugation of N n over F . Proposition 3.3.
The group N n normalizes each term of the flag { T n, , . . . , T n,n − } .Proof. It is enough to prove that each element of N n \ U n normalizes T n,i for 0 ≤ i ≤ n − g ∈ N n \ U n , from [ACGS19, Corollary 3] we have that T n,n − = T n ∩ T gn is normalin N n . Hence, for every subgroup H = T n,i where i < n − g ∈ N n \ U n , wehave H g ≤ T n,n − . If x ∈ U n , we clearly have ( H g ) x = ( H gxg − ) g = H g . Since T n is a uniserial U -module, we conclude that H g belongs to F n . Thus T gn,i = H g = T n,i . (cid:3) We also used
GAP to calculate N kn for n ≤
11. The computational results are summarized inFig. 1, where the entry in position ( k, n ) denotes the logarithm in base 2 of the size of N k − n . Weobserve that, in each column, consecutive values above the diagonal (bold values in the figure)have fixed differences. Such differences are listed in the (auxiliary) last column. For example, thenumber “+7” appearing in the last position of the fifth row denotes that the difference betweenlog | N j | and log | N j | equals 7, where 5 ≤ j ≤
11, reading the table from left to right, startingfrom the position (5 ,
5) containing the bold number, i.e. the number 35.n 2 3 4 5 6 7 8 9 10 11log | U n | | N n | -
11 16 22 29 37 46 56 67 +1log | N n | - -
18 24 31 39 48 58 69 +2log | N n | - - 14
28 35 43 52 62 73 +4log | N n | - - 15 23
42 50 59 69 80 +7log | N n | - - - 25 37
61 70 80 91 +11log | N n | - - - 27 41 57
86 96 107 +16log | N ,n | - - - 28 45 64 84
119 130 +23log | N n | - - - 29 46 67 89 113
162 +32log | N n | - - - 30 47 71 95 122 155 +43 Figure 1.
The logarithm of the size of the normalizers, when n ≤ (cid:12)(cid:12) N kn : N k − n (cid:12)(cid:12) , reported in the last column of Fig. 1,do not depend on n , if n ≥ k + 2, and match with those of the sequence { a j } j ≥ of the partialsums of the sequence { b j } j ≥ counting the number of partitions of j into at least two dis-tinct parts. The reader is referred to The On-Line Encyclopedia of Integer Sequences at [OEI, https://oeis.org/A317910 ] for a list of values and additional information. In the next section
CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 6 j b j a j Figure 2.
First values of the sequences a j and b j we show that for small values of k this is actually true. The above evidence is now summarizedhere as a conjecture. Conjecture 1.
For n ≥ k + 2 ≥ , the number log (cid:12)(cid:12) N kn : N k − n (cid:12)(cid:12) is independent of n and is equalto ( k + 2) -th term of the sequence { a j } j ≥ of the partial sums of the sequence { b j } j ≥ countingthe number of partitions of j into at least two distinct parts. The first values of the sequences a j and b j are listed in Fig. 2.4. Theoretical evidence
In this section we prove Theorem 4.7 which solves Conjecture 1 in the cases 1 ≤ k ≤
4, byproviding an explicit construction of N kn . We first need the following general lemma. Lemma 4.1.
Let G = A ⋊ B be a group and H a subgroup of G containing B . If [ N A ( H ∩ A ) , B ] ≤ H , then N G ( H ) = N A ( H ∩ A ) ⋊ B. Proof.
Clearly B ≤ H ≤ N G ( H ). Let x ∈ N A ( H ∩ A ) ≤ A E G . Then [ H, x ] ⊆ A , since A is normal in G . Let h ∈ H and let us write h = bk where b ∈ B and k ∈ A ∩ H . We have[ h, x ] = [ bk, x ] = [ b, x ] k [ k, x ] ∈ H , since [ b, x ] ∈ H as [ N A ( H ∩ A ) , B ] ≤ H by hypotheses. Thus[ H, x ] ⊆ H ∩ A and thus N A ( H ∩ A ) ≤ N G ( H ).Let x ∈ N A ( H ∩ A ), k ∈ H ∩ A and b ∈ B . Notice that k x b = (( k b − |{z} ∈ H ∩ A ) x ) b ∈ H ∩ A. This implies that N A ( H ∩ A ) is normalized by B . As a consequence, we have N G ( H ) ≥ N A ( H ∩ A ) ⋊ B. Conversely, let x ∈ N G ( H ). Since G = A ⋊ B , we can find b ∈ B ≤ H ≤ N G ( H ) such that x = bu with u ∈ A . Clearly, u ∈ N G ( H ) ∩ A = N A ( H ). If h ∈ H ∩ A , then [ u, h ] ∈ H ∩ A , since A is normal in G . Thus x = bu ∈ N A ( H ∩ A ) ⋊ B . (cid:3) Consider now the set-wise stabilizer Q n in Σ n of X n − . This group acts also on X s n n − = { n − + 1 , . . . , n } and so Q n = Σ n − × (Σ n − ) s n and Σ n = Q n ⋊ h s n i , where s n interchangesthe two direct factors of Q n . We can give a general procedure for the construction of the nor-malizer N Σ n ( Y ) of a subgroup Y ≤ Σ n containing T n such that [ N Q n ( Y ∩ Q n ) , s n ] ⊆ Y . Since t n = s n ∈ Y , we have Y s n = Y and N Σ n ( Y ) = N Q n ( Y ∩ Q n ) ⋊ h s n i by Lemma 4.1.Let us apply the previous construction to obtain a description of U n as the normalizer of T n in Σ n . Proposition 4.2.
We have that U n = h s n i ⋉ (cid:0) ∆ n ( U n − ) · ( T n − × T s n n − ) (cid:1) = h s n i ⋉ (cid:0) ∆ n ( U n − ) · T n − (cid:1) . CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 7 Proof.
Using the same notation as above, we notice that T n ∩ Q n = ∆ n ( T n − ) = { ( t, t s n ) | t ∈ T n − } . We first claim that N Q n ( T n ∩ Q n ) = ∆ n ( U n − ) · T n − . It is straightforward to check that ∆ n ( U n − ) normalizes T n ∩ Q n = ∆ n ( T n − ) and that T n − centralizes T n ∩ Q n , hence N Q n ( T n ∩ Q n ) ≥ ∆ n ( U n − ) · T n − . Now, let x = ( a, b s n ) ∈ N Q n ( T n ∩ Q n ), where a, b ∈ Σ n − , and y = ( t, t s n ) ∈ ∆ n ( T n − ) = T n ∩ Q n < T n − × T s n n − . We have that y x = ( t a , ( t b ) s n ) = (¯ t, ¯ t s n ) ∈ ∆ n ( T n − ) < T n − × T s n n − for some ¯ t ∈ T n − , and so t a = ¯ t = t b . It follows that a, b ∈ U n − and ab − ∈ C Σ n − ( T n − ) = T n − by Remark 1. Therefore a = b ˜ t , with ˜ t ∈ T n − and x = ( b, b s n ) · (˜ t, ∈ ∆ n ( U n − ) · T n − ,giving the opposite inclusion. In conclusion, N Q n ( T n ∩ Q n ) = ∆ n ( U n − ) · T n − . In order to apply Lemma 4.1, it remains to be shown that [∆ n ( T n − ) · T n − , s n ] ≤ T n . Noticethat ∆ n ( T n − ) · T n − = T n − × T s n n − , and thus [∆ n ( T n − ) · T n − , s n ] ≤ ∆ n ( T n − ) ≤ T n , asclaimed. (cid:3) Proposition 4.3.
Let H E K ≤ Σ n − and U def = h s n i ⋉ (∆ n ( K ) · ( H × H s n )) . If we define • L def = N Σ n − ( K ) ∩ N Σ n − ( H ) , • M def = C K ( K/H ) ,then N Σ n ( U ) = h s n i ⋉ (∆ n ( L ) · ( M × M s n )) . Moreover, M E L ≤ Σ n − .Proof. The inclusion N Σ n ( U ) ≥ h s n i ⋉ (∆ n ( L ) · ( M × M s n )) is straightforward since every factorof the second member is contained in the first one.Note that U ∩ Q n = ∆ n ( K ) · ( H × H s n ). Let us start considering the group N def = N Q n ( U ) = N Σ n ( U ) ∩ Q n . Let x = ( a, b s n ) ∈ N , where a, b ∈ Σ n − . First we note that [ x, s n ] = ( a − b, ( b − a ) s n ) ∈ U ∩ Q n = ∆ n ( K ) · ( H × H s n ). In particular, a − b ∈ K .Let y = ( h, s n ) ∈ H × H s n ≤ U ∩ Q n , where h ∈ H and 1 ∈ Σ n − . We have y x = ( h a , s n ) ∈ ∆ n ( K ) · ( H × H s n ) = ∆ n ( K ) ⋉ ( H × n ( K ) ∩ ( H ×
1) = 1, then h a ∈ H for all h ∈ H andso a ∈ N Σ n − ( H ). Similarly, we have that b ∈ N Σ n − ( H ). Now, letting u = ( k, k s n ) ∈ ∆ n ( K ),we have u x = (cid:0) k a , ( k b ) s n (cid:1) = ( k a , ( k a ) s n ) · (cid:0) , (( k a ) − k b ) s n (cid:1) (1) ∈ ∆ n ( K ) · ( H × H s n )= ∆ n ( K ) ⋉ (1 × H s n ) , and so a ∈ N Σ n − ( K ) ∩ N Σ n − ( H ) = L . Similarly, b ∈ N Σ n − ( K ) ∩ N Σ n − ( H ) = L . Again byEq. (1) we have b = am with m = a − b ∈ C L ( K/H ) ∩ K = M . It follows that x = ( a, b s n ) = ( a, a s n ) · (1 , m s n ) ∈ ∆ n ( L ) · ( M × M s n ) . CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 8 Hence N ≤ ∆ n ( L ) · ( M × M s n ). As a consequence we have N Σ n ( U ) = h s n i ⋉ N = h s n i ⋉ (cid:0) ∆ n ( L ) · ( M × M s n ) (cid:1) , as required.We also trivially have that M ≤ L ≤ Σ n − . If m ∈ M , k ∈ K and l ∈ L then[ k, m l ] = l − [ k l − , m ] | {z } ∈ H l ∈ H, and therefore m l ∈ M and M E L . (cid:3) The following technical definition is necessary to provide a recursive construction for thenormalizer chain of T n in Σ n . Definition 4.4.
For a given natural number n we define the series (cid:8) C kn (cid:9) k ≥ and (cid:8) D kn (cid:9) k ≥ recursively as follows: C n def = T n ,D n def = N Σ n ( C n ) = U n ,C kn def = C D k − n (cid:0) D k − n /C k − n (cid:1) for k ≥ ,D kn def = N Σ n ( C k − n ) ∩ N Σ n ( D k − n ) for k ≥ . Proposition 4.5.
For each k ≥ we have that N kn = h s n i ⋉ (cid:0) ∆ n ( D kn − ) · (cid:0) C kn − × ( C kn − ) s n (cid:1)(cid:1) = h s n i ⋉ (cid:0) ∆ n ( D kn − ) ⋉ (cid:0) C kn − × { } (cid:1)(cid:1) . Proof.
The result follows by a recursive application of Proposition 4.3, assuming H = C k − n − , K = D k − n − , L = D kn − and M = C kn − , beginning with C n − = T n − which is normal in D n − = U n − . (cid:3) The case ≤ k ≤ . The main result of this work will be proved in this section. In orderto do so, let us denote by Θ n the group of the upper unitriangular matrices and by Z h (Θ n ) the h -th term of its upper central series.By Proposition 4.2 U n = h s n i ⋉ (cid:0) ∆ n ( U n − ) ⋉ ( T n − × { } ) (cid:1) and T n = h s n i · ∆ n ( T n − ) . Hence Θ n ∼ = U n /T n = ∆ n ( U n − /T n − ) ⋉ ( T n − × { } ) = ∆ n (Θ n − ) ⋉ ( T n − × { } ) . Moreover, notice that Θ n = U n − . It is easily checked that Z (Θ n ) = T n − , × { } . Proceeding by induction we obtain the following generalization.
Lemma 4.6.
We have that Z h (Θ n ) = ∆ n ( Z h − (Θ n − )) ⋉ ( T n − ,h × { } ) . Proof. If G h def = ∆ n ( Z h − (Θ n − )) ⋉ ( T n − ,h × { } ), then G h /G h − is a central section of Θ n ,hence G h ≤ Z h (Θ n ). Notice that | G h : G h − | = | Z h (Θ n ) : Z h − (Θ n ) | , which is known to be 2 h .Therefore Z h (Θ n ) = G h . (cid:3) CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 9 We are now ready to prove our main result.
Theorem 4.7.
Let n be a non-negative integer. Then Conjecture 1 is true for ≤ k ≤ .Proof. Let us prove each case separately. We will use Proposition 4.3 repeatedly without furthermention. Since C n = T n and D n = U n , by Proposition 4.2 we have N n = h s n i ⋉ (cid:0) ∆ n ( U n − ) ⋉ ( T n − × { } ) (cid:1) = U n . [ k = ] Since C n = C U n ( U n /T n ) = Z (Θ n ) ⋉ T n and D n = U n ∩ N Σ n ( U n ) = U n , we obtain N n = h s n i ⋉ (cid:0) ∆ n ( U n − ) ⋉ (( Z (Θ n − ) ⋉ T n − ) × { } ) (cid:1) , and so (cid:12)(cid:12) N n : N n (cid:12)(cid:12) = (cid:12)(cid:12) N n : U n (cid:12)(cid:12) = 2 = 2 , since | Z (Θ n − ) | = 2.[ k = ] We have C n = C U n (cid:0) U n / ( Z (Θ n ) ⋉ T n ) (cid:1) = Z (Θ n ) ⋉ T n and D n = N Σ n ( C n ) ∩ N Σ n ( U n )= N Σ n ( Z (Θ n ) ⋉ T n ) ∩ N n = N N n ( Z (Θ n ) ⋉ T n )= U n . The last equality depends on the fact that | T n · T gn | = 2 n +2 , where g ∈ N n \ U n from [ACGS19, Corollary 3], and that | ( Z (Θ n ) ⋉ T n ) | = 2 n +1 . Weconsequently obtain that N n = h s n i ⋉ (cid:0) ∆ n ( U n − ) ⋉ (( Z (Θ n − ) ⋉ T n − ) × { } ) (cid:1) , and so (cid:12)(cid:12) N n : N n (cid:12)(cid:12) = 2 = 4, since | Z (Θ n − ) : Z (Θ n − ) | = 2 .[ k = ] We have that C n = C U n ( U n / (cid:0) Z (Θ n ) ⋉ T n ) (cid:1) = Z (Θ n ) ⋉ T n and D n = N Σ n ( Z (Θ n ) ⋉ T n ) ∩ N Σ n ( U n )= N Σ n ( Z (Θ n ) ⋉ T n ) ∩ N n = N N n ( Z (Θ n ) ⋉ T n )= N n . In order to prove last equality, we first show that [ N n , T n ] ≤ Z (Θ n ) ⋉ T n . For each g ∈ N n \ U n , we have that [[ T gn , U n ] , U n ] is a normal subgroup of U n of index at least4 in T gn . Since T gn is uniserial for U n , then [[ T gn , U n ] , U n ] is necessarily contained in theunique subgroup of index 4 in T gn and normal in U n , which is T gn ∩ T n (see [ACGS19]).Hence ( T n · T gn ) /T n lies in the second term of the upper central series of the quotient U n /T n = Θ n . Thus T gn ≤ Z (Θ n ) ⋉ T n . We are left with proving that [ N n , Z (Θ n )] ≤ Z (Θ n ) ⋉ T n , which is a direct consequence of the following straightforward properties: • Z (Θ n ) = ∆ n ( Z (Θ n − )) ⋉ ( T n − , × { } ) (Lemma 4.6); • [ s n , ∆ n ( Z (Θ n − )) ⋉ ( T n − , × { } )] ≤ T n ; • [∆ n ( U n − ) , ∆ n ( Z (Θ n − ))] ≤ T n ; • [∆ n ( U n − ) , T n − , × { } ] ≤ T n − , × { } ≤ Z (Θ n ); • [ Z (Θ n − ) ⋉ T n − , ∆ n ( Z (Θ n − ))] ≤ T n − , × { } = Z (Θ n ) ≤ Z (Θ n ); • [ Z (Θ n − ) ⋉ T n − , T n − , × { } ] = { } .In conclusion, we derive that N n = h s n i ⋉ (cid:0) ∆ n ( N n − ) ⋉ (( Z (Θ n − ) ⋉ T n − ) × { } ) (cid:1) , CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 10 and so (cid:12)(cid:12) N n : N n (cid:12)(cid:12) = 2 = 16, as (cid:12)(cid:12) N n − : U n − (cid:12)(cid:12) = 2 and | Z (Θ n − ) : Z (Θ n − ) | = 2 .The same result can be also obtained as follows. Note that Z (Θ n ) ⋉ T n = Z ( U n ) · T n .Since Z ( U n ) is a characteristic subgroup of U n we have ( Z (Θ n ) ⋉ T n ) x = ( Z ( U n ) · T n ) x = Z ( U n ) · T xn ≤ Z (Θ n ) · T n · T xn = Z (Θ n ) ⋉ T n for all x ∈ N n . As a consequence[ N n , Z (Θ n )] ≤ [ N n , Z (Θ n ) T n ] ≤ Z (Θ n ) ⋉ T n .[ k = ] We mimic the argument provided for the case k = 3. We start by computing C n = C N n (cid:0) N n / ( Z (Θ n ) ⋉ T n ) (cid:1) = C M ∆ n (Θ n − ) ⋉ (cid:0) ( Z (Θ n − ) ⋉ T n − ) × { } (cid:1) ∆ n ( Z (Θ n − )) ⋉ (cid:0) ( Z (Θ n − ) ⋉ T n − , ) × { } (cid:1) ! ⋉ T n = (cid:0) ∆ n ( Z (Θ n − )) ⋉ (cid:0) ( Z (Θ n − ) ⋉ T n − , ) × { } (cid:1)(cid:1) ⋉ T n , where M def = ∆ n (Θ n − ) ⋉ (cid:0) ( Z (Θ n − ) ⋉ T n − ) × { } (cid:1) . Moreover D n = N Σ n ( Z (Θ n ) ⋉ T n ) ∩ N Σ n ( N n )= N Σ n ( Z (Θ n ) ⋉ T n ) ∩ N n = N N n ( Z (Θ n ) ⋉ T n )= N n . The last equality is derived proceeding as in the case k = 3, provided that n is sufficientlylarge (e.g. n ≥ N n = h s n i ⋉ (cid:0) ∆ n ( N n − ) ⋉ (cid:0) C n − × { } (cid:1)(cid:1) , and so (cid:12)(cid:12) N n : N n (cid:12)(cid:12) = 2 . Indeed, (cid:12)(cid:12) N n − : N n − (cid:12)(cid:12) = 2 and (cid:12)(cid:12) C n − : C n − (cid:12)(cid:12) = 2 , since C n − = (cid:0) ∆ n − ( Z (Θ n − )) ⋉ (cid:0) ( Z (Θ n − ) ⋉ T n − , ) × { } (cid:1)(cid:1) ⋉ T n − and C n − = (cid:0) ∆ n − ( Z (Θ n − )) ⋉ ( T n − , × { } ) (cid:1) ⋉ T n − , where | Z (Θ n − ) : Z (Θ n − ) | = 2 , | T n − , , T n − , | = 2 and | Z (Θ n − ) | = 2.Finally, notice that, if n ≥ k + 2, the construction of N kn described above does not depend onthe dimension n and the integers corresponding to log (cid:12)(cid:12) N kn : N k − n (cid:12)(cid:12) for 1 ≤ k ≤ { a j } of the partial sums of thesequence { b j } counting the number of partitions of j into at least two distinct parts (see Fig. 2and the auxiliary column of Fig. 1). Therefore, Conjecture 1 is true for 1 ≤ k ≤ (cid:3) Although we have a recursive method, it appears that the construction of N kn requires ad hoc computations that become increasingly complex as k grows. Open Problem 1.
Find a closed and concise formula for N kn . Moreover, even though the sequence of the indices log (cid:12)(cid:12) N kn : N k − n (cid:12)(cid:12) seems to be predictablefor k ≤ n −
2, as conjectured in this paper, it is hard to figure any conjecture on the valuesappearing under the bold diagonal of the table in Fig. 1.
Open Problem 2.
Determine log (cid:12)(cid:12) N kn : N k − n (cid:12)(cid:12) for all natural numbers k and n . CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 11 Some computational aspects
We can derive from Proposition 4.2 the following efficient construction of U n , which has beenuseful to speed up the process of computing the normalizer chain.The center Z ( U n ) is the subgroup h t n i , which is actually the center of Σ n . Let u n ,j def = t nn − j +1 for 1 ≤ j ≤ n, u n ,j def = u n − ,j − for 2 ≤ j ≤ n, and for 3 ≤ i ≤ j ≤ n u ni,j def = u n − i − ,j − ( u n − i − ,j − ) s n . Using this notation, it easy to recognize that U = T ,U = (cid:10) u , , u , , u , (cid:11) ,U = (cid:10) u , , u , , u , , u , , u , , u , (cid:11) , ... U n = (cid:10) u ni,j | ≤ i ≤ j ≤ n (cid:11) . As an example of this construction, we conclude the paper by showing the
GAP code whichwe used to build the normalizer chains displayed in Fig. 1. The orders of the normalizer are alsoprovided. The code below is specialized to the case n = 8. dim:=8;gens:=[]; CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 12 Append(newngens,gens);od;newgens:=Set(newgens);newngens:=Set(newngens);gens:=newgens;ngens:=newngens;Add(gens, x);Add(ngens,x);Add(sgens,x);tmpsyl:=Group(ngens);ngens:=MinimalGeneratingSet(tmpsyl);tmpsyl:=false;od;t:=Group(gens);
Acknowledgment
We are thankful to the staff of the Department of Information Engineering, Computer Scienceand Mathematics at the University of L’Aquila for helping us in managing the HPC clusterCALIBAN, which we extensively used to run our simulations ( caliband.disim.univaq.it ). We
CHAIN OF NORMALIZERS IN THE SYLOW 2-SUBGROUPS OF SYM(2 n ) 13 are also grateful to the Istituto Nazionale d’Alta Matematica - F. Severi for regularly hostingour research seminar
Gruppi al Centro in which this paper was conceived.
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