Counterexample to a conjecture about braces

Structure of a group and the structure of its lattice of subgroups

We classify nilpotent associative algebras of dimensions up to 4 over any field. This is done by constructing the nilpotent associative algebras as central extensions of algebras of smaller dimension, analogous to methods known for nilpotent Lie algebras.

Classification of nilpotent associative algebras of small dimension

As interest in pro-p Groupss has grown, so has interest in the Nottingham group. Known to number theorists as the group of wild automorphisms of the local field$ \mathcal{N} = \mathcal{N}({\mathbb{F}_q})$ (where q = p e ), this finitely generated pro-p Groups.$ \mathcal{N} = \mathcal{N}({\mathbb{F}_q})$ was introduced to the group theory community in the work of D. Johnson [13] (himself inspired by an article of S. Jennings [12]) and his Ph.D. student I. York [26] [27]. Viewing S as a group of formal power series under substitution, D. Johnson and I. York proved many of the initial structural results. Some of the proofs are reproduced below and the reader will see the ease with which one can calculate within the group. This, alongside the group’s interesting properties, has led to the Nottingham group becoming a favourite test case for conjectures concerning pro-p Groupss.

The Nottingham Group

We study graded Lie algebras of maximal class over a field $ \mathbf {F}$ of positive characteristic $p$. A. Shalev has constructed infinitely many pairwise non-isomorphic insoluble algebras of this kind, thus showing that these algebras are more complicated than might be suggested by considering only associated Lie algebras of p-groups of maximal class. Here we construct $| \mathbf {F}|^{\aleph _{0}}$ pairwise non-isomorphic such algebras, and $\max \{| \mathbf {F}|, \aleph _{0} \}$ soluble ones. Both numbers are shown to be best possible. We also exhibit classes of examples with a non-periodic structure. As in the case of groups, two-step centralizers play an important role

/pdf/graded-lie-algebras-of-maximal-class-4vpvgpyglb.pdf

Graded Lie Algebras of Maximal Class

The main theorem of this paper is that if (N, +) is a finite abelian p-group of p-rank m where m + 1 < p, then every regular abelian subgroup of the holomorph of N is isomorphic to N . The proof utilizes a connection, observed in [CDVS06], between regular abelian subgroups of the holomorph of N and nilpotent ring structures on (N, +). Examples are given that limit possible generalizations of the theorem. The primary application of the theorem is to Hopf Galois extensions of fields. Let L|K be a Galois extension of fields with abelian Galois group G. If also L|K is HHopf Galois where the K-Hopf algebra H has associated group N with N as above, then N is isomorphic to G.

/pdf/abelian-hopf-galois-structures-on-prime-power-galois-field-10r6q376dh.pdf

Abelian Hopf Galois structures on prime-power Galois field extensions

The group generated by the round functions of a block cipher has been widely investigated. We identify a large class of block ciphers for which this group is easily guaranteed to be primitive. Our class includes the AES cipher and the SERPENT cipher.

Andrea Caranti

Papers

Graded Lie Algebras of Maximal Class

Abelian Hopf Galois structures on prime-power Galois field extensions

On some block ciphers and imprimitive groups

Graded Lie algebras of maximal class, II

Abelian regular subgroups of the affine group and radical rings