Mal’tsev bases for partially commutative nilpotent groups

We construct an ordered set of commutators in a partially commutative nilpotent group [Formula: see text]. This set allows us to define a canonical form for each element of [Formula: see text]. Namely, we construct a Mal’tsev basis for the group [Formula: see text]

On the primitive irreducible representations of finitely generated nilpotent groups

We develop some tecniques whish allow us to apply the methods of commutative algebra for studing the representations of nilpotent groups. Using these methods, in particular, we show that any irreducible representation of a finitely generated nilpotent group G over a finitely generated field of characteristic zero is induced from a primitive representation of some subgroup of G.

Diameters of random Cayley graphs of finite nilpotent groups

We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.

The Non-Commuting, Non-Generating Graph of a Nilpotent Group

For a nilpotent group $G$, let $\Xi(G)$ be the difference between the complement of the generating graph of $G$ and the commuting graph of $G$, with vertices corresponding to central elements of $G$ removed. That is, $\Xi(G)$ has vertex set $G \setminus Z(G)$, with two vertices adjacent if and only if they do not commute and do not generate $G$. Additionally, let $\Xi^+(G)$ be the subgraph of $\Xi(G)$ induced by its non-isolated vertices. We show that if $\Xi(G)$ has an edge, then $\Xi^+(G)$ is connected with diameter $2$ or $3$, with $\Xi(G) = \Xi^+(G)$ in the diameter $3$ case. In the infinite case, our results apply more generally, to any group with every maximal subgroup normal. When $G$ is finite, we explore the relationship between the structures of $G$ and $\Xi(G)$ in more detail.

On permutable strongly 2-maximal and strongly 3-maximal subgroups

We describe the structure of finite solvable non-nilpotent groups in which every two strongly n-maximal subgroups are permutable (n = 2; 3). In particular, it is shown for a solvable non-nilpotent group G that any two strongly 2-maximal subgroups are permutable if and only if G is a Schmidt group with Abelian Sylow subgroups. We also prove the equivalence of the structure of non-nilpotent solvable groups with permutable 3-maximal subgroups and with permutable strongly 3-maximal subgroups. The last result allows us to classify all finite solvable groups with permutable strongly 3-maximal subgroups, and we describe 14 classes of groups with this property. The obtained results also prove the nilpotency of a finite solvable group with permutable strongly n -maximal subgroups if the number of prime divisors of the order of this group strictly exceeds n (n=2; 3).

Cryptographic multilinear maps using pro-p groups

<p style='text-indent:20px;'>In [<xref ref-type="bibr" rid="b18">18</xref>], the authors show how, to any nilpotent group of class <inline-formula><tex-math id="M2">\begin{document}$ n $\end{document}</tex-math></inline-formula>, one can associate a non-interactive key exchange protocol between <inline-formula><tex-math id="M3">\begin{document}$ n+1 $\end{document}</tex-math></inline-formula> users. The <i>multilinear</i> commutator maps associated to nilpotent groups play a key role in this protocol. In the present paper, we explore some alternative platforms, such as pro-<inline-formula><tex-math id="M4">\begin{document}$ p $\end{document}</tex-math></inline-formula> groups.</p>

An answer to a conjecture on the sum of element orders

Let [Formula: see text] be a finite group and [Formula: see text], where [Formula: see text] denotes the order of [Formula: see text]. The function [Formula: see text] was introduced by Tărnăuceanu. In [M. Tărnăuceanu, Detecting structural properties of finite groups by the sum of element orders, Israel J. Math. (2020), https://doi.org/10.1007/s11856-020-2033-9 ], some lower bounds for [Formula: see text] are determined such that if [Formula: see text] is greater than each of them, then [Formula: see text] is cyclic, abelian, nilpotent, supersolvable and solvable. Also, an open problem aroused about finite groups [Formula: see text] such that [Formula: see text] is equal to the amount of each lower bound. In this paper, we give an answer to the equality condition which is a partial answer to the open problem posed by Tărnăuceanu. Also, in [M. Baniasad Azad and B. Khosravi, A criterion for p-nilpotency and p-closedness by the sum of element orders, Commun. Algebra (2020), https://doi.org/10.1080/00927872.2020.1788571 ], it is shown that: If [Formula: see text], where [Formula: see text] is a prime number, then [Formula: see text] and [Formula: see text] is cyclic. As the next result, we show that if [Formula: see text] is not a [Formula: see text]-nilpotent group and [Formula: see text], then [Formula: see text].

Nilpotence relations in products of groups

Two subgroups 𝐴 and 𝐵 of a group 𝐺 are said to be 𝒩-connected if, for all 𝑎 in 𝐴 and 𝑏 in 𝐵, the subgroup generated by 𝑎 and 𝑏 is a nilpotent group. In this paper, we study the structure of a group 𝐺 assuming that G=AB and 𝐴 and 𝐵 are 𝒩-connected subgroups satisfying Max or Min.

Growth rate for endomorphisms of finitely generated nilpotent groups

We prove that the growth rate of an endomorphism of a finitely generated nilpotent group is equal to the growth rate of the induced endomorphism on its abelianization, generalizing the corresponding result for an automorphism in [T. Koberda, Entropy of automorphisms, homology and the intrinsic polynomial structure of nilpotent groups, In the Tradition of Ahlfors–Bers. VI, Contemp. Math. 590, American Mathematical Society, Providence 2013, 87–99].