Rigid commutators and a normalizer chain

Abstract

The notion of rigid commutators is introduced to determine the sequence of the logarithms of the indices of a certain normalizer chain in the Sylow 2-subgroup of the symmetric group on $$2^n$$ 2 n letters. The terms of this sequence are proved to be those of the partial sums of the partitions of an integer into at least two distinct parts, that relates to a famous Euler’s partition theorem.

Verification and generation of unrefinable partitions

“…Unrefinable partitions have been introduced by Aragona et al [ACGS22] where they have appeared in a natural way in connection to some subgroups in a chain of normalizers [ACGS21]. It is proved, in particular, that such generators are represented by some unrefinable partitions satisfying conditions on the minimal excludant.…”

A modular idealizer chain and unrefinability of partitions with repeated parts

“…Aragona et al [ACGS21b] have recently shown that, for 1 ≤ i ≤ n − 2, a transversal of N i−1 in N i can be put in one-to-one correspondence with a set of partitions into distinct parts in such a way that, denoting by {q 2,i } i≥1 the partial sum of the sequence {p 2,i } i≥1 of partitions into distinct parts, the following equality is satisfied:…”

“…In a subsequent work [ACGS22], the authors introduced the concept of unrefinable partitions and proved that a transversal of N n−2 in N n−1 is in one-toone correspondence with a set of unrefinable partitions whose minimal excludant satisfies an additional requirement. The study of the chain on normalizers (N i ) i≥0 has been carried out up to the (n−1)-th term by means of rigid commutators [ACGS21b], a set of generators of Σ, which is closed under commutation and which was intentionally designed for the purpose. However, the technique of rigid commutators could not be easily generalized to the odd case of the normalizer chain, i.e., the one defined in a Sylow p-subgroup of Sym(p n ), with p odd.…”

Unrefinable partitions into distinct parts in a normalizer chain

“…The sequence (b j ) of the number of partitions of integers into distinct parts has been extensively studied in past and recent years and is a well-understood integer sequence [Eul48,And94] appearing in different areas of mathematics. Recently, triggered by a problem in algebraic cryptography related to translation subgroups in the symmetric group with 2 n elements [CDVS06, ACGS19, CBS19, CCS21], it has been shown that such a sequence is related to the growth of the indices of consecutive terms in a chain of normalizers [ACGS21]. More precisely, let Σ n be a Sylow 2-subgroup of the symmetric group Sym(2 n ) and T be an elementary abelian regular subgroup of Σ n .…”

“…generated by rigid commutators, a family of left-normed commutators involving a special set of generators of Σ n . We invite the interested reader to refer to Aragona et al [ACGS21], where rigid commutators and saturated groups are introduced and described in detail. When i > n−2, the behavior of the chain does not seem to show any recognizable pattern and the study of its combinatorial nature is still open.…”