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

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 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.