Related papers: Lie rings related to the $p$-groups of maximal cla…

We classify the nilpotent Lie rings of order $p^8$ with maximal class for $p \ge 5$. This also provides a classification of the groups of order $p^8$ with maximal class for $p \ge 11$ via the Lazard correspondence.

We develop a Lazard correspondence between post-Lie rings and skew braces that satisfy a natural completeness condition. This is done through a thorough study of how the Lazard correspondence behaves on semi-direct sums of Lie rings. In…

For a prime p and natural number n with p greater than or equal to n, we establish the existence of a non-functorial one-to-one correspondence between isomorphism classes of groups of order p^n whose derived subgroup has exponent dividing…

We study Lie rings definable in a finite-dimensional theory, extending the results for the finite Morley rank case. In particular, we prove a classification of Lie rings of dimension up to four in the NIP or connected case. In…

The graph G(p) associated with the p-groups of maximal class is a major tool in their classification. We introduce a subgraph of the graph G(p) called its frame. Its construction is based on the Lazard correspondence. We show that every…

In this paper a simple algebraic formula is obtained for the correspondence between finite right nilpotent Fp-braces and finite nilpotent pre-Lie algebras. This correspondence agrees with the correspondence using Lazard's correspondence…

Let p be a prime number. We show that there is a one-to-one correspondence between the set of strongly nilpotent braces and the set of nilpotent pre-Lie rings of cardinality $p^{n}$, for sufficiently large p. Moreover, there is an injective…

In this paper, several theorems of Macdonald \cite{Mac1961,Mac1962} on the varieties of nilpotent groups will be generalized to the case of Lie rings. We consider three varieties of Lie rings of any characteristic associated with some…

The class of elementary totally disconnected groups is the smallest class of totally disconnected, locally compact, second countable groups which contains all discrete countable groups, all metrizable pro-finite groups, and is closed under…

Let $p>3$ be a prime number and let $A$ be a brace whose additive group is a direct sum of cyclic groups of cardinalities larger than $p^{\alpha }$ for some $\alpha $. Suppose that either (i) $A^{\lfloor{\frac {p-1}4}\rfloor}\subseteq pA$…

We investigate coherency properties of certain completed integral group rings, precisely for compact $p$-adic Lie groups.

We study nilpotent groups acting faithfully on complex algebraic varieties. We use a method of base change. For finite p-groups, we go from $k$, a number field, to a finite field in order to use counting lemmas. We show that a finite…

Like the lower central series of a nilpotent group, filters generalize the connection between nilpotent groups and graded Lie rings. However, unlike the case with the lower central series, the associated graded Lie ring may share few…

We prove nilpotency results for Lie algebras over an arbitrary field admitting a derivation, which satisfies a given polynomial identity $r(t)=0$. For the polynomial $r=t^n-1$ we obtain results on the nilpotency of Lie algebras admitting a…

We give a systematic study of the model theory of generic nilpotent groups and Lie algebras. We show that the Fra\"iss\'e limit of 2-nilpotent groups of exponent $p$ studied by Baudisch is 2-dependent and NSOP$_{1}$. We prove that the class…

Lazard correspondence provides an isomorphism of categories between finitely generated nilpotent pro-$p$ groups of nilpotency class smaller than $p$ and finitely generated nilpotent $\mathbb{Z}_p$-Lie algebras of nilpotency class smaller…

We establish an equivalence between categories of 'formally nilpotent' Lie algebras and exponential groups in characteristic zero. It extends the equivalences of Mal'cev, Lazard, Quillen and Warfield, and applies to groups under composition…

Lazard showed in his seminal work "Groupes analytiques $p$-adiques" that for rational coefficients continuous group cohomology of $p$-adic Lie-groups is isomorphic to Lie-algebra cohomology. We refine this result in two directions: firstly…

In this note we, first, recall that the sets of all representatives of some special ordinary residue classes become $\left( m,n\right) $-rings. Second, we introduce a possible $p$-adic analog of the residue class modulo a $p$-adic integer.…

We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in [1], in terms of Igusa functions. As corollaries we obtain information about…