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 \({{\,\mathrm{AGL}\,}}(2,n)\), 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 coincides with the sequence of the partial sums of the numbers of partitions into at least two distinct parts.
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Acknowledgements
We are thankful to the staff of the Department of Information Engineering, Computer Science and Mathematics at the University of L’Aquila for helping us in managing the HPC cluster CALIBAN, which we extensively used to run our simulations (caliband.disim.univaq.it). We are also grateful to the Istituto Nazionale d’Alta Matematica - F. Severi for regularly hosting our research seminar Gruppi al Centro in which this paper was conceived. Finally, we thank the referee for some useful remarks.
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Communicated by Gadadhar Misra.
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. N. Gavioli is a member of the Centre of Excellence EX-EMERGE at University of L’Aquila.
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Aragona, R., Civino, R., Gavioli, N. et al. A chain of normalizers in the sylow \(\mathbf{2}\)-subgroups of the symmetric group on \({\mathbf{2}}^{\varvec{n}}\) letters. Indian J Pure Appl Math 52, 735–746 (2021). https://doi.org/10.1007/s13226-021-00190-w
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DOI: https://doi.org/10.1007/s13226-021-00190-w