Nilpotent Groups Of Class Research Articles

On decidability of the product of subgroups membership problem for nilpotent groups

Abstract The product of subgroups membership problem for a finitely generated group 𝐺 is the decision problem, where, given two or more finitely generated subgroups K i K_{i} , i = 1 , … , m i=1,\ldots,m , of 𝐺 and a group element 𝑔, it is asked whether g ∈ ∏ i = 1 m K i g\in\prod_{i=1}^{m}K_{i} . In this paper, we prove that, for any finitely generated nilpotent group 𝑁 of class two, the product of two subgroups membership problem is decidable. Note that this problem is equivalent to the problem of determining the non-emptiness of the intersection of two cosets of finitely generated subgroups. We also show that, for a sufficiently large direct power H n \mathbb{H}^{n} of the Heisenberg group ℍ, there exists a product K = ∏ i = 1 4 K i K=\prod_{i=1}^{4}K_{i} of four finitely generated subgroups for which the membership problem is algorithmically undecidable. The proof of the last assertion is based on the undecidability of Hilbert’s 10th problem and interpretation of Diophantine equations in nilpotent groups. The question of decidability of the membership problem for a product of three finitely generated subgroups in a nilpotent group of class two remains open.

Groups of nilpotency class 2 of order p4 as additive groups of local nearrings

Groups of nilpotency class 2 of order p4 as additive groups of local nearrings

A note on the bound for the class of certain nilpotent groups

Assume G is a nilpotent group of class > 3 in which every proper subgroup has class at most 3. In this note, we give the exact upper bound of class of G.

On the isomorphism problem for split extensions

In this paper, we study an important well-known structure operation, namely, the semidirect product (or split extensions) which is a very useful tool to structure certain kinds of groups. More precisely, we study the isomorphism problem for semidirect products and then we determine how isomorphism of semidirect products and conjugacy of the images of the corresponding actions are related. As an application, for two positive integers [Formula: see text] and [Formula: see text], we compute the number of upper isomorphism classes of split extensions of an elementary abelian [Formula: see text]-group of order [Formula: see text] by an elementary abelian [Formula: see text]-group of order [Formula: see text]. Furthermore, we deal with split extensions where the kernel is a non-abelian [Formula: see text]-group of nilpotency class two and the quotient is an elementary abelian [Formula: see text]-group.

Finite p-groups which are non-inner nilpotent

A group G is called a non-inner nilpotent group, whenever it is nilpotent with respect to a non-inner automorphism. In 2018, all finitely generated abelian non-inner nilpotent groups have been classified. Actually, the authors proved that a finitely generated abelian group G is a non-inner nilpotent group, if G is not isomorphic to cyclic groups Z_p_1p_2...p_t and Z, for a positive integer t and distinct primes p_1, p_2,..., p_t. We conjecture that all finite non-abelian p-groups are non-inner nilpotent and we prove this conjecture for finite $p$-groups of nilpotency class 2 or of co-class 2.

Noether's Problem for Abelian Extensions of Cyclic p-groups of Nilpotency Class 2

Let $$K$$ be a field and $$G$$ be a finite group. Let $$G$$ act on the rational function field $$K(x(g):g\in G)$$ by $$K$$-automorphisms defined by $$g\cdot x(h)=x(gh)$$ for any $$g,h\in G$$. Denote by $$K(G)$$ the fixed field $$K(x(g):g\in G)^G$$. Noether's problem then asks whether $$K(G)$$ is rational over $$K$$. Let $$p$$ be prime and let $$G$$ be a $$p$$-group of exponent $$p^e$$. Assume also that (i) $$K = p>0$$, or (ii) $$K \ne p$$ and $$K$$ contains a primitive $$p^e$$-th root of unity. In this paper we prove that $$K(G)$$ is rational over $$K$$ if $$G$$ is any finite $$p$$-group of nilpotency class $$2$$ which is an abelian extension of a cyclic group.

Fibonacci length and the generalized order k-Pell sequences of the 2-generator p-groups of nilpotency class 2

In this paper, we consider the 2-generator p-groups of nilpotency class 2. We find the Fibonacci length and the period of the generalized order k-Pell sequences of these groups.

Factorization of the $t$-extension of the $p$-Fibonacci and the Pascal matrices

In this paper, we consider the 2-generator $p$-groups of nilpotency class 2. We find the Fibonacci length and the period of the generalized order $k$--Pell sequences of these groups.

Weakly totally permutable products and Fitting classes

It is known that if $ G=AB $ is a product of its totally permutable subgroups $ A $ and $ B $‎, ‎then $ Gin mathfrak{F} $ if and only if $ Ain mathfrak{F} $ and $ Bin mathfrak{F} $ when $ mathfrak{F} $ is a Fischer class containing the class $ mathfrak{U} $ of supersoluble groups‎. ‎We show that this holds when $ G=AB $ is a weakly totally permutable product for a particular Fischer class‎, ‎$ mathfrak{F}diamond mathfrak{N} $‎, ‎where $ mathfrak{F} $ is a Fitting class containing the class $ mathfrak{U} $ and $ mathfrak{N} $ a class of nilpotent groups‎. ‎We also extend some results concerning the $ mathfrak{U} $-hypercentre of a totally permutable product to a weakly totally permutable product‎.

The Axiomatic Rank of the Levi Class Generated by the Almost Abelian Quasivariety of Nilpotent Groups

Let $$\mathcal{M}$$ be a quasivariety generated by a relatively free group in a class of nilpotent groups of step $$\leq 2$$ with commutator subgroups of prime exponent $$p,$$ $$p\neq 2.$$ A class $$L(\mathcal{M})$$ of all groups $$G$$ the normal closure of any element in which belongs to $$\mathcal{M}$$ is called the Levi class generated by $$\mathcal{M}.$$ It is proved that $$L(\mathcal{M})$$ has finite axiomatic rank, i.e., $$L(\mathcal{M})$$ can be defined by a system of quasi-identities with a finite number of variables.

Word problems for finite nilpotent groups

Let w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that N_w(1)ge |G|^{k-1}, where for gin G, the quantity N_w(g) is the number of k-tuples (g_1,ldots ,g_k)in G^{(k)} such that w(g_1,ldots ,g_k)={g}. Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that N_w(g)ge |G|^{k-1} for g a w-value in G, and prove that N_w(g)ge |G|^{k-2} for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

On the triple tensor product of prime-power groups

Abstract Let G be a finite p-group, and let ⊗ 3 G {\otimes^{3}G} be its triple tensor product. In this paper, we obtain an upper bound for the order of ⊗ 3 G {\otimes^{3}G} , which sharpens the bound given by G. Ellis and A. McDermott, [Tensor products of prime-power groups, J. Pure Appl. Algebra 132 1998, 2, 119–128]. In particular, when G has a derived subgroup of order at most p, we classify those groups G for which the bound is attained. Furthermore, by improvement of a result about the exponent of ⊗ 3 G {\otimes^{3}G} determined by G. Ellis [On the relation between upper central quotients and lower central series of a group, Trans. Amer. Math. Soc. 353 2001, 10, 4219–4234], we show that, when G is a nilpotent group of class at most 4, exp ( ⊗ 3 G ) {\exp(\otimes^{3}G)} divides exp ⁡ ( G ) {\exp(G)} .

Homology of Free Nilpotent Lie Rings

Some results obtained by the calculations of the integral homology of free nilpotent Lie algebras Hi (L(x1, x2,. . . , xr)/γN+1) in the system for computational discrete algebra GAP are presented. The attention is focused on the occurrence of unexpected torsion in this homology, similar to one that arises for a 4-generated free nilpotent group of class 2. The main result is that even for the case of two generators, the torsion occurs in the fourth integral homology when the nilpotency class is 5. Moreover, only 7-torsion occurs, and no others. Namely, there is an isomorphism H4(L(x1, x2)/γ6) ≅ ℤ85 ⨁ ℤ /7ℤ. We also give an explicit mathematical proof of the fact that H4(L(x1, x2, x3, x4)/γ3) has nontrivial 3-torsion. Bibliography: 6 titles.

A Levi Class Generated by a Quasivariety of Nilpotent Groups

Let L(M) be a class of all groups G in which the normal closure of any element belongs to M; qM is a quasivariety generated by a class M. We consider a quasivariety qH2 generated by a relatively free group in a class of nilpotent groups of class at most 2 with commutator subgroup of exponent 2. It is proved that the Levi class L(qH2) generated by the quasivariety qH2 is contained in the variety of nilpotent groups of class at most 3.

Low-complexity computations for nilpotent subgroup problems

We solve the following algorithmic problems using [Formula: see text] 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.

Equality of derival automorphisms and certain automorphisms in finitely generated groups

Let [Formula: see text] be a group and [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] denote the group of all inner automorphisms, the group of all pointwise inner automorphisms, the group of all central automorphisms and the group of all derival automorphisms of [Formula: see text], respectively. We know that in a finite [Formula: see text]-group [Formula: see text] of class 2, [Formula: see text] if and only if [Formula: see text] is cyclic and [Formula: see text], where [Formula: see text] is the group of all derival automorphisms of [Formula: see text] which fix [Formula: see text] elementwise. In this paper, we characterize all finite nilpotent groups of class 2 for which [Formula: see text] or [Formula: see text] is equal to [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. Also, we characterize all finitely generated nilpotent groups of class 2 for which [Formula: see text] is equal to [Formula: see text] and give some interesting corollaries in this regard.

A 5-Engel associative algebra whose group of units is not 5-Engel

Let R be an associative ring with unity and let [R] and U(R) denote the associated Lie ring (with [a,b]=ab−ba) and the group of units of R, respectively. In 1983 Gupta and Levin proved that if [R] is a nilpotent Lie ring of class c then U(R) is a nilpotent group of class at most c. The aim of the present note is to show that, in general, a similar statement does not hold if [R] is n-Engel. We construct an algebra R over a field of characteristic ≠2,3 such that the Lie algebra [R] is 5-Engel but the group U(R) is not.

Construction of Right Gyrogroups

Attempts have been made to introduce notion of class assigned functions k for a given group G and observed that it determines a right gyrogroup $$(G, o_k)$$ . It is also observes that $$(G, o_k)$$ will be a group if and only if G is a nilpotent group of class 2 and will be a gyrogroup for a nilpotent group of class 3, where $$k(x)=1$$ for all $$x\in G\setminus \{1\}$$ .

Universal input of knapsack problem for groups

In this work we investigate the group version of the well known knapsack problem (KP) in class of nilpotent groups. We show that there exists the universal input for this problem. In other word it mean that there is the two step nilpotent group Gu and parametric input In, where n is natural parameter, such that for any torsion free nilpotent group G and input I for KP there exists a natural number m such that KP for group Gu on input Im is equivalent to KP for group G on input I.

On Dominions of the Rationals in Nilpotent Groups

The dominion of a subgroup H of a group G in a class M is the set of all a ∈ G that have the same images under every pair of homomorphisms, coinciding on H from G to a group in M. A group H is n-closed in M if for every group G = gr(H, a1,..., an) in M that includes H and is generated modulo H by some n elements, the dominion of H in G (in M) is equal to H. We prove that the additive group of the rationals is 2-closed in every quasivariety of torsion-free nilpotent groups of class at most 3.