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

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.