School of Mathematics & Physics, University of Lincoln
On Wednesday the 10th of November 2021, Norberto Gavioli (Università degli Studi dell’Aquila) will be giving an online seminar on his research at 4.45pm. The details of his talk are as follows:
Title: A chain of normalizers in the Sylow 2-subgroup of Sym(2^n).
Abstract: The normalizer of a regular elementary abelian 2-subgroup in the symmetric group Sym(2^n) is the affine group AGL(2^n). Due to some questions arising from differential cryptanalysis of block cyphers, we started studying the normalizer chain in Sym(2^n) starting from the Sylow 2-subgroup of AGL(2^n). Computational results suggested a possible connection with the sequence (a_m) of the number of partitions of m into distinct parts already studied by Euler. We could prove indeed that for i\le n-2 the the base 2 logarithm index of the i-th term of the chain in the (i-1)-th one is equal to (i-2)-th term of the partial sum sequence arising from (a_m). This result rely on a combinatorial structure, that will be described, involving a family of commutators in a suitable set of generators of the Sylow 2-subgroup of Sym(2^n).