Narrow Results

We consider mutually permutable products G = AB of two nilpotent groups. The structure of the Sylow p-subgroups of its nilpotent residual is described.

We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x₀, x₁, …, x_N ∈ G, N ∈ ℕ, then one can always find a uniformly computably enumerable (i.e. uniformly ) ascendant sequence of order type ω + 1 of subgroups in G beginning with 〈x₀, x₁, …, x_N〉_G, the subgroup generated by x₀, x₁, …, x_N in G. This complexity is surprisingly low in light of the fact that the usual definition of ascendant sequence involves arbitrarily large ordinals that index sequences of subgroups defined via a transfinite recursion in which each step is incomputable. We produce this surprisingly low complexity sequence via the effective algebraic commutator collection process of P. Hall, and a related purely algebraic Normal Form Theorem of M. Hall for nilpotent groups.

This paper studies nilpotent and Hamiltonian cancellative residuated lattices and their relationship with nilpotent and Hamiltonian lattice-ordered groups. In particular, results about lattice-ordered groups are extended to the domain of residuated lattices. The two key ingredients that underlie the considerations of this paper are the categorical equivalence between Ore residuated lattices and lattice-ordered groups endowed with a suitable modal operator; and Malcev’s description of nilpotent groups of a given nilpotency class c in terms of a semigroup equation.

It is well-known that no knot can be cancelled in a connected sum with another knot, whereas every link can be cancelled up to link homotopy in a (componentwise) connected sum with another link. In this paper we address the question whether the noncancellation property of knots holds for (piecewise-linear) links up to some stronger analogue of link homotopy, which still does not distinguish between sufficiently close C⁰-approximations of a topological link. We introduce a sequence of such increasingly stronger equivalence relations under the name of k-quasi-isotopy, k∈ℕ; all of them are weaker than isotopy (in the sense of Milnor). We prove that every link can be cancelled up to peripheral structure preserving isomorphism of any quotient of the fundamental group, functorially invariant under k-quasi-isotopy; functoriality means that the isomorphism between the quotients for links related by any allowable crossing change fits in the commutative diagram with the fundamental group of the complement to the intermediate singular link. The proof invokes Baer's theorem on the join of subnormal locally nilpotent subgroups. On the other hand, the integral generalized (lk ≠ 0) Sato–Levine invariant is invariant under 1-quasi-isotopy, but is not determined by any quotient of the fundamental group (endowed with the peripheral structure), functorially invariant under 1-quasi-isotopy — in contrast to Waldhausen's theorem.

As a byproduct, we use to determine the image of the Kirk–Koschorke invariant of fibered link maps.

Let G be a group and let Aut_c(G) be the group of all central automorphisms of G. Let C* = C_Autc(G)(Z(G)) be the set of all central automorphisms of G fixing Z(G) elementwise. In this paper, we prove that if G is a finitely generated nilpotent group of class 2, then C* ≃ Inn(G) if and only if Z(G) is cyclic or Z(G) ≃ C_m × ℤ^r where has exponent dividing m and r is torsion-free rank of Z(G). Also we prove that if G is a finitely generated group which is not torsion-free, then C* = Inn(G) if and only if G is nilpotent group of class 2 and Z(G) is cyclic or Z(G) ≃ C_m × ℤ^r where has exponent dividing m and r is torsion-free rank of Z(G). In both cases, we show G has a particularly simple form.

Let 𝔛 be a class of groups. A group G is said to be minimal non-𝔛 if all proper subgroups of G are 𝔛-groups but G itself is not. The aim of this paper is to study the class of minimal non-FCⁿ-groups, where FCⁿ (n is a positive integer) is a class of generalized FC-groups introduced in [F. de Giovanni, A. Russo and G. Vincenzi, Groups with restricted conjugacy classes, Serdica Math. J.28 (2002) 241–254].

Let G be a finite group and e a positive integer dividing |G|, the order of G. Denoting L_e(G) = {x ∈ G|x^e = 1}. Frobenius proved that |L_e(G)| = ke for some positive integer k ≥ 1. Let k(G) be the upper bound of the set {k||L_e(G)| = ke, ∀ e ||G|}. In this paper, a complete classification of the finite group G with k(G) = 3 is obtained.

Let G be a group and let M and N be two normal subgroups of G. Let be the set of all automorphisms of G which centralize G/M and N. In this paper, we find certain necessary and sufficient conditions on G such that be equal to Z(Inn(G)), Inn(G), C* or Aut_c(G). We also characterize the subgroups of a finite p-group G for which the equality holds.

An automorphism of a group G is called an IA-automorphism if it induces the identity automorphism on the abelianized group G/G′. Let IA(G) denote the group of all IA-automorphisms of G. We classify all finitely generated nilpotent groups G of class 2 for which IA(G) ≃ Inn(G). In particular, we classify all finite nilpotent groups of class 2 for which each IA-automorphism is inner.

Let G be a group and Z_j(G), for j ≥ 0, be the jth term in the upper central series of G. We prove that if Out^c(G/Z_j(G)), the group of outer class-preserving automorphisms of G/Z_j(G), is nilpotent of class k, then Out^c(G) is nilpotent of class at most j + k. Moreover, if Out^c(G/Z_j(G)) is a trivial group, then Out^c(G) is nilpotent of class at most j. As an application we prove that if γ_i(G)/γ_i(G) ∩ Z_j(G) is cyclic then Out^c(G) is nilpotent of class at most i + j, where γ_i(G), for i ≥ 1, denotes the ith term in the lower central series of G. This extends an earlier work of the author, where this assertion was proved for j = 0. We also improve bound on the nilpotency class of Out^c(G) for some classes of nilpotent groups G.

Let $G$ be a finite group. If $K$ is a subgroup of $G$ and $H$ a subgroup of $K$, we say that $H$ is strongly closed in $K$ with respect to $G$ if $K\cap H^g\le H$ for any $g\in G$. We say that a subgroup $H$ of $G$ is strongly closed in $G$ if $H$ is strongly closed in $N_G(H)$ with respect to $G$. A subgroup $H$ of a group $G$ is said to be weakly $c$-supplemented in $G$ if $G$ has a subgroup $K$ such that $G=HK$ and $H\cap K$ is strongly closed in $G$. In this paper, we study the structure of a group $G$ under the assumption, that some subgroups of prime power order are weakly $c$-supplemented in $G$. Our results extend and generalize several recent results in the literature.

An element $a$ of a ring $R$ is nil-clean, if $a=e+b$, where $e^2=e\in R$ and $b$ is a nilpotent element, and the ring $R$ is called nil-clean if each of its elements is nil-clean. In [W. Wm. McGovern, S. Raja and A. Sharp, Commutative nil clean group rings, J. Algebra Appl.14(6) (2015) 5; Article ID: 1550094], it was proved that, for a commutative ring $R$ and an abelian group $G$, the group ring $RG$ is nil-clean, iff $R$ is nil-clean and $G$ is a $2$-group. Here, we discuss the nil-cleanness of group rings in general situation. We prove that the group ring of a locally finite $2$-group over a nil-clean ring is nil-clean, and that the hypercenter of the group $G$ must be a $2$-group if a group ring of $G$ is nil-clean. Consequently, the group ring of a nilpotent group over an arbitrary ring is nil-clean, iff the ring is a nil-clean ring and the group is a $2$-group.

Let $G$ be any group and $G$ be a subgroup of $\mathrm{\text{Sym}}(\mathrm{\Omega })$ for some set $\mathrm{\Omega }$. The $2$-closure of $G$ on $\mathrm{\Omega }$, denoted by $G^{(2),\mathrm{\Omega }}$, is by definition,

Let $G$ be a group and $\mathrm{\text{Inn}}(G)$, ${\mathrm{\text{Aut}}}_{pwi}(G)$, ${\mathrm{\text{Aut}}}_c(G)$ and $D(G)$ 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 $G$, respectively. We know that in a finite $p$-group $G$ of class 2, $D(G)=\mathrm{\text{Inn}}(G)$ if and only if $G^{\prime }$ is cyclic and $D(G)=D^{\ast }(G)$, where $D^{\ast }(G)$ is the group of all derival automorphisms of $G$ which fix $Z(G)$ elementwise. In this paper, we characterize all finite nilpotent groups of class 2 for which $D(G)$ or $D^{\ast }(G)$ is equal to $\mathrm{\text{Inn}}(G)$, $C^{\ast }(G)$, ${\mathrm{\text{Aut}}}_c(G)$ and ${\mathrm{\text{Aut}}}_{pwi}(G)$. Also, we characterize all finitely generated nilpotent groups of class 2 for which $D^{\ast }(G)$ is equal to $\mathrm{\text{Inn}}(G)$ and give some interesting corollaries in this regard.

In this paper, we prove that a group is nilpotent of class at most two if and only if each of its self-centralizing subgroups is normal. We also show that if all self-centralizing subgroups of a certain group are subnormal then all subgroups are subnormal.

The purpose of this paper is to introduce a concept of commutator in a multiplicative Lie algebra which will be termed as a Lie commutator. Consequently, we develop the theory of Lie solvability and Lie nilpotency of multiplicative Lie algebras.

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.

Let m and n be positive integer numbers. In this paper, we study $𝔑(m,n)$, the class of all groups G that for all subsets m and n of G containing m and n elements, respectively, there exist $x\in M$ and $y\in N$ such that $〈x,y〉$ is nilpotent, which introduced by Zarrin in 2012. We improve some results of Zarrin and find some sharp bounds for m and n such that $G\in 𝔑(m,n)$ implies that G is nilpotent. Also we will characterize all finite $𝔫$-groups in $𝔑(m,n)$, which $m+n\le 10$.

Let $G$ be a finite nilpotent group. We prove the following results. (1) If $G$ is of class 2 and acts faithfully and irreducibly on an elementary abelian group $V$, then all nontrivial orbits of $G$ on $V$ have sizes larger than $|G{|}^{1/2}$. (2) If $G^{\prime }$ is cyclic, then every subgroup of $G$ intersecting trivially with the center of $G$ has order less than $|G{|}^{1/2}$. We also show that a result like (2) cannot be obtained when the hypothesis that $G^{\prime }$ is cyclic is replaced by the hypothesis that the center of $G$ is cyclic.

Let G be a finite group. A subgroup H of G is called conjugate-permutable in G if HH^g = H^gH for any g ∈ G. A group G is called an ECP-group if every subgroup of G is conjugate-permutable in G. In this paper, we study the influence of conjugate-permutable subgroups on the structure of a finite group, especially on the nilpotency or supersolvability of the group, and give some sufficient or necessary conditions for a finite group to be an ECP-group.