Regular subgroups with large intersection

In this paper we study the relationships between the elementary abelian regular subgroups and the Sylow $2$-subgroups of their normalisers in the symmetric group $\mathrm{Sym}(\mathbb{F}_2^n)$, in view of the interest that they have recently raised for their applications in symmetric cryptography

A modular idealizer chain and unrefinability of partitions with repeated parts

Recently Aragona et al. have introduced a chain of normalizers in a Sylow 2-subgroup of Sym(2^n), starting from an elementary abelian regular subgroup. They have shown that the indices of consecutive groups in the chain depend on the number of partitions into distinct parts and have given a description, by means of rigid commutators, of the first n-2 terms in the chain. Moreover, they proved that the (n-1)-th term of the chain is described by means of rigid commutators corresponding to unrefinable partitions into distinct parts. Although the mentioned chain can be defined in a Sylow p-subgroup of Sym(p^n), for p > 2 computing the chain of normalizers becomes a challenging task, in the absence of a suitable notion of rigid commutators. This problem is addressed here from an alternative point of view. We propose a more general framework for the normalizer chain, defining a chain of idealizers in a Lie ring over Z_m whose elements are represented by integer partitions. We show how the corresponding idealizers are generated by subsets of partitions into at most m-1 parts and we conjecture that the idealizer chain grows as the normalizer chain in the symmetric group. As an evidence of this, we establish a correspondence between the two constructions in the case m=2

The breadth-degree type of a finite p-group

In the present paper we show that a stem finite p-group G has size bounded by min(p^(8d 122log2d+b 124)(b+1)/2,p^b(3b+4d 121)/2) where b is the breadth of G and pd is the maximum character degree of G. As a consequence there are only finitely many finite stem p-groups having breadth b and maximum character degree pd

A Chain of Normalizers in the Sylow 22-subgroups of the symmetric group on 2n2^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 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

Rigid commutators and a normalizer chain

The novel 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 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

GGS-groups over primary trees: Branch structures

We study branch structures in Grigorchuk-Gupta-Sidki groups (GGS-groups) over primary trees, that is, regular rooted trees of degree $p^n$ for a prime $p$. Apart from a small set of exceptions for $p=2$, we prove that all these groups are weakly regular branch over $G''$. Furthermore, in most cases they are actually regular branch over $\gamma_3(G)$. This is a significant extension of previously known results regarding periodic GGS-groups over primary trees and general GGS-groups in the case $n=1$. We also show that, as in the case $n=1$, a GGS-group generated by a constant vector is not branch.Comment: 14 page

Ideally rr-constrained graded Lie subalgebras of maximal class algebras

Let $E\supseteq F$ be a field extension and $M$ a graded Lie algebra of maximal class over $E$. We investigate the $F$-subalgebras $L$ of $M$, generated by elements of degree $1$. We provide conditions for $L$ being either ideally $r$-constrained or not just infinite. We show by an example that those conditions are tight. Furthermore, we determine the structure of $L$ when the field extension $E\supseteq F$ is finite. A class of ideally $r$-constrained Lie algebras which are not $(r-1)$-constrained is explicitly constructed, for every $r\geq 1$