A chain of normalizers in the sylow $$\mathbf{2}$$ 2 -subgroups of the symmetric group on $${\mathbf{2}}^{\varvec{n}}$$ 2 n letters

Riccardo Aragona (), Roberto Civino (), Norberto Gavioli () and Carlo Maria Scoppola ()
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Riccardo Aragona: Università degli Studi dell’Aquila
Roberto Civino: Università degli Studi dell’Aquila
Norberto Gavioli: Università degli Studi dell’Aquila
Carlo Maria Scoppola: Università degli Studi dell’Aquila

Indian Journal of Pure and Applied Mathematics, 2021, vol. 52, issue 3, 735-746

Abstract: Abstract 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 $${{\,\mathrm{AGL}\,}}(2,n)$$ AGL ( 2 , n ) , each term of the chain is defined as the normalizer of the previous one in the symmetric group on $$2^n$$ 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 coincides with the sequence of the partial sums of the numbers of partitions into at least two distinct parts.

Keywords: Symmetric group on $$2^n$$ 2 n elements; Elementary abelian regular subgroups; Sylow 2-subgroups; Normalizers; 20B30; 20B35; 20D20 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s13226-021-00190-w

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