Regular subgroups with large intersection

Aragona, Riccardo; Civino, Roberto; Gavioli, Norberto; Scoppola, Carlo Maria

doi:10.1007/s10231-019-00853-w

Cited by 12 publications

(9 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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 [4] it is proved that U n contains, as normal subgroups, exactly two conjugates of T , namely T and T U n = 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.…”

Section: Elementary Abelian Regular 2-groups and Their Chain Of Normamentioning

confidence: 99%

“…. 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 [15], 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 [4,9,10]. This is relevant in the cryptanalysis of block ciphers, since it may trigger a variation of the well-known differential attack [7]: a statistical attack which allows us 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.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Rigid commutators and a normalizer chain

et al. 2021

Self Cite

View full text Add to dashboard Cite

The 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 $$2^n$$ 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.

show abstract

Section: Elementary Abelian Regular 2-groups and Their Chain Of Normamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Rigid commutators and a normalizer chain

et al. 2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…U n ] is necessarily contained in the unique subgroup of index 4 in T g n and normal in U n , which is T g n ∩ T n (see [ACGS19]). Hence (T n • T g n )/T n lies in the second term of the upper central series of the quotient…”

Section: Theoretical Evidencementioning

confidence: 99%

“…In a recent paper [ACGS19], we considered the elements of the conjugacy class T Sym(2 n ) which are subgroups of the affine group AGL(T ). We showed that, if T g ∩ T has index 4 in T , then there exists a Sylow 2-subgroup U < AGL(T ) containing both T g and T as normal subgroups.…”

Section: Introductionmentioning

confidence: 99%

A Chain of Normalizers in the Sylow $2$-subgroups of the symmetric group on $2^n$ letters

Aragona,

Civino,

Gavioli

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

On the basis of an initial interest in symmetric cryptography, in the present work we study a chain of subgroups. Starting from a Sylow 2-subgroup of AGL(2, n), each term of 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 large values 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 the partial sums of the numbers of partitions into at least two distinct parts. 2010 Mathematics Subject Classification. 20B30, 20B35, 20D20. Key words and phrases. Symmetric group on 2 n elements; Elementary abelian regular subgroups; Sylow 2-subgroups; Normalizers. All the authors are members of INdAM-GNSAGA (Italy). R. Civino is partially funded by the Centre of excellence ExEMERGE at University of L'Aquila. Part of this work has been carried out during the cycle of seminars "Gruppi al Centro" organized at INdAM in Rome.

show abstract

“…Integer partitions into distinct parts may appear in several areas of mathematics, sometimes unexpectedly. For example, they have been recently shown to be linked to the set of generators of groups in a group-theoretical problem related to cryptography [ACGS19,ACGS21a]. In particular, Aragona et al showed that the generators of a given group are linked to partitions into distinct parts which satisfy a condition of non-refinability [ACGS21b] together with a condition on the minimal excludant.…”

Section: Introductionmentioning

confidence: 99%

On the maximal part in unrefinable partitions of triangular numbers

Aragona¹,

Campioni²,

Civino³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

A partition into distinct parts is refinable if one of its parts a can be replaced by two different integers which do not belong to the partition and whose sum is a, and it is unrefinable otherwise. Clearly, the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distributions of the parts. We prove a O(n 1/2 )-upper bound for the largest part in an unrefinable partition of n, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions for triangular numbers, proving that if n is even there exists only one maximal unrefinable partition of n(n + 1)/2, and that if n is odd the number of such partitions equals the number of partitions of ⌈n/2⌉ into distinct parts. In the second case, an explicit bijection is provided.

show abstract

Regular subgroups with large intersection

Cited by 12 publications

References 17 publications

Rigid commutators and a normalizer chain

Rigid commutators and a normalizer chain

A Chain of Normalizers in the Sylow $2$-subgroups of the symmetric group on $2^n$ letters

On the maximal part in unrefinable partitions of triangular numbers

Contact Info

Product

Resources

About