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Bera [Line graph characterization of power graphs of finite nilpotent groups, Comm. Algebra50(11) (2022) 4652–4668] characterized finite nilpotent groups whose power graphs and proper power graphs are line graphs. In this paper, we extend the results of above-mentioned paper to arbitrary finite groups. Also, we correct the corresponding result of the proper power graphs of dihedral groups. Moreover, we classify all the finite groups whose enhanced power graphs are line graphs. We classify all the finite nilpotent groups (except non-abelian $2$-groups) whose proper enhanced power graphs are line graphs of some graphs. Finally, we determine all the finite groups whose (proper) power graphs and (proper) enhanced power graphs are the complement of line graphs, respectively.

We consider descending HNN extensions $〈B,t:t^{-1}Ht=K〉$, where $B=H$ is abelian. We show that class-preserving automorphisms of descending HNN extensions are all inner if $B$ is abelian and $K\ne B$. It follows that outer automorphism groups of such conjugacy separable descending HNN extensions are residually finite.

Let $G$ be a finite nilpotent group. We denote by $\gamma (G)$ the number of conjugacy classes of noncyclic subgroups of $G$, $\mathrm{\text{Con}}(G)$ the number of conjugacy classes of subgroups of $G$. In this paper, we give the formula for $\gamma (G)$, and prove that $\mathrm{\text{Con}}(G)-\gamma (G)\ge 4$ when the order of $G$ is prime power. Moreover, we get the minimum of $\gamma (G)$ according to $t$, where $t$ is the number of distinct noncyclic Sylow subgroups of $G$.

In this paper, we are interested in the relationships between the structure of a finite group $G$ and the zeros of strongly monolithic characters of $G$. We give some criteria for solvability and supersolvability of finite groups.

Let N = V ⊕ Z be a step-two nilpotent group. We study the Radon transform on functions on N defined by integration over the orbits {(z, t)(v, 0); v ∈ V} of the points (z, t) by the action of the subspace V ⊕ 0 of N. For N being of H-type, Siegel-type defined as the Shilov boundary of a complex Siegel domain, or Type F_{2r, b}, a real analogue of the Siegel type, we find an inversion and a Plancherel formula for the Radon transform. We prove certain L^p - L^q boundedness properties for the transform. This generalizes earlier works of Geller–Stein and Strichartz for the Heisenberg groups.

We investigate the possibility of discrete groups furnishing a kinematic framework for systems whose thermodynamic behavior may be given by nonadditive entropies. Relying on the well-known result of the growth rate of balls of nilpotent groups, we see that maintaining extensivity of the entropy of a nilpotent group requires using a non-Boltzmann/Gibbs/Shannon (BGS) entropic form. We use the Tsallis entropy as an indicative alternative. Using basic results from hyperbolic and random groups, we investigate the genericity and possible range of applicability of the BGS entropy in this context. We propose a sufficient condition for phase transitions, in the context of (multi-) parameter families of nonadditive entropies.

Questions about nilpotency of groups satisfying Engel conditions have been considered since 1936, when Zorn proved that finite Engel groups are nilpotent. We prove that 4-Engel groups are locally nilpotent. Our proof makes substantial use of both hand and machine calculations.

We let G be a group, and we let k be a natural number. We assume that G is nilpotent of class at most k, and that every (k + 1)-ary congruence preserving function on G is a polynomial function.

We show that then every congruence preserving function on G (of any finite arity) is a polynomial function.

There is a simple group-theoretic formula for the second integral homology group of a group. This is an abelian group and there is an analogous formula for another abelian group, which involves a normal subgroup N of a torsion-free nilpotent group G. Properties of this abelian group translate into properties of G/N. This approach allows one to give a simple purely group-theoretic proof of an old theorem of J. R. Stallings, namely that if Γ is a group, if H₁(G,ℤ) is free abelian and if H₂(G,ℤ) = 0, then any subset Y of G which is independent modulo the derived group of G, freely generates a free group. The ideas used admit to considerable generalization, yielding in particular, proofs of a number of theorems of U. Stammbach.

The breadth of a polycyclic group is the maximum of h(G) - h(C_G(x)) for x ∈ G, where h(G) is the Hirsch length. We prove a number of results that bound the class of a finitely generated nilpotent group, and the Hirsch length of the derived group in a polycyclic group, in terms of the breadth. These results are analogues of well-known results in finite group theory.

This paper is motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ⁰ of all finitely generated free algebras in Θ. The key problem is to study how far the group AutΘ⁰ of all automorphisms of the category Θ⁰ is from the group InnΘ⁰ of inner automorphisms of Θ⁰ (see [7, 10] for details). Recall that an automorphism ϒ of a category 𝔎 is inner, if it is isomorphic as a functor to the identity automorphism of the category 𝔎.

Let Θ = 𝔑_d be the variety of all nilpotent groups whose nilpotency class is ≤ d. Using the method of verbal operations developed in [8, 9], we prove that every automorphism of the category , d ≥ 2 is inner.

We describe a nilpotent quotient algorithm for a certain class of infinite presentations: the so-called finite L-presentations. We then exhibit finite L-presentations for various examples and report on the application of our nilpotent quotient algorithm to them. As a result, we obtain conjectural descriptions of the lower central series structure of various interesting groups including the Grigorchuk supergroup, the Brunner–Sidki–Vieira group, the Basilica group, certain generalizations of the Fabrykowski–Gupta group, and certain generalizations of the Gupta–Sidki group.

We established several new criteria for existence of complements and supplements to some normal abelian subgroups in groups. In passing, as one of the many useful applications and corollaries of these results, we obtained a description of some finitely generated soluble groups of finite Hirsch–Zaitsev rank. As another application of our results, we obtained a D.J.S. Robinson's theorem on structure of finitely generated soluble groups of finite section rank. The original proof of this theorem was homological, but all proofs in this paper, including this one, are purely group-theoretical.

We describe an algorithm for deciding whether or not a given finitely generated torsion-free nilpotent group is decomposable as the direct product of nontrivial subgroups.

In this paper, we study distortion of various well-known embeddings of finitely generated torsion-free nilpotent groups $G$ into unitriangular groups $U{T}_n(\mathbb{Z})$. In particular, we show that there is no undistorted embeddings of $5$-dimensional Heisenberg group into $U{T}_n(\mathbb{Z})$. We also provide a polynomial time algorithm for finding distortion of a given subgroup of $G$.

Let $A$ be a nilpotent $p$-group of finite exponent and $B$ be an abelian $p$-group of finite exponent for a given prime number $p$. Then the wreath product $A\mathrm{\text{Wr}}B$ generates the variety $\mathrm{\text{var}}(A)\mathrm{\text{var}}(B)$ if and only if the group $B$ contains a subgroup isomorphic to the direct product $C_{p^v}^{\infty }$ of countably many copies of the cycle $C_{p^v}$ of order $p^v=\mathrm{\text{exp}}(B)$. The obtained theorem continues our previous study of cases when $\mathrm{\text{var}}(A\mathrm{\text{Wr}}B)=\mathrm{\text{var}}(A)\mathrm{\text{var}}(B)$ holds for some other classes of groups $A$ and $B$ (abelian groups, finite groups, etc.).

In this paper, we prove that certain HNN extensions of finitely generated abelian subgroup separable groups are finitely generated abelian subgroup separable. Using this, we show that certain HNN extensions of finitely generated nilpotent groups are finitely generated abelian subgroup separable.

We solve the following algorithmic problems using $T{C}^0$ circuits, or in logspace and quasilinear time, uniformly in the class of nilpotent groups with bounded nilpotency class and rank: subgroup conjugacy, computing the normalizer and isolator of a subgroup, coset intersection, and computing the torsion subgroup. Additionally, if any input words are provided in compressed form as straight-line programs or in Mal’cev coordinates, the algorithms run in quartic time.

In this paper, we use the notion of twisted subgroups (i.e. subsets of group elements closed under the binary operation $(a,b)\mapsto aba$) to provide the first structural characterization of optimal play in the Explorer-Director game, introduced as the Magnus–Derek game by Nedev and Muthukrishnan and generalized to finite groups by Gerbner. In particular, we reduce the game to the problem of finding the largest proper twisted subgroup, and as a corollary we resolve the Explorer-Director game completely for all nilpotent groups.

Recently, Macdonald et al. showed that many algorithmic problems for finitely generated nilpotent groups including computation of normal forms, the subgroup membership problem, the conjugacy problem, and computation of subgroup presentations can be done in $\text{LOGSPACE}$. Here, we follow their approach and show that all these problems are complete for the uniform circuit class ${TC}^0$ — even if an $r$-generated nilpotent group of class at most $c$ is part of the input but $r$ and $c$ are fixed constants. In particular, unary encoded systems of a bounded number of linear equations over the integers can be solved in ${TC}^0$. In order to solve these problems in ${TC}^0$, we show that the unary version of the extended gcd problem (compute greatest common divisors and express them as linear combinations) is in ${TC}^0$. Moreover, if we allow a certain binary representation of the inputs, then the word problem and computation of normal forms is still in uniform ${TC}^0$, while all the other problems we examine are shown to be ${TC}^0$-Turing-reducible to the binary extended gcd problem.