Narrow Results

Let $X$ be a projective variety of dimension $n\ge 1$ over an algebraically closed field of arbitrary characteristic. We prove a Fujiki–Lieberman type theorem on the structure of the automorphism group of $X$. Let $G$ be a group of zero entropy automorphisms of $X$ and $G_0$ the set of elements in $G$ which are isotopic to the identity. We show that after replacing $G$ by a suitable finite-index subgroup, $G/G_0$ is a unipotent group of the derived length at most $n-1$. This result was first proved by Dinh et al. for compact Kähler manifolds.

Let K be a field of odd characteristic p, and let G be the direct product of a finite p-group P ≠ 1 and a Hamiltonian 2-group. We show that the set of symmetric elements (KG)_* of the group algebra KG with respect to the involution of KG which inverts all elements of G, satisfies all Lie commutator identities of degree t(P) or more, where t(P) denotes the nilpotency index of the augmentation ideal of the group algebra KP. In addition, if P is powerful, then (KG)_* satisfies no Lie commutator identity of degree less than t(P). Applying this result we get that (KG)_* is Lie nilpotent and Lie solvable, and its Lie nilpotency index and Lie derived length are not greater than t(P) and ⌈log₂ t(P)⌉, respectively, and these bounds are attained whenever P is a powerful group. The corresponding result on the set of symmetric units of KG is also obtained.

For any group G, let denote the set of centralizers of G. We say that a group G is a -group if . We prove that every arbitrary -group is center-by-(finite of order . Also we show that the derived length of a nilpotent -group is at most [n - 1/2] + 1.

We study the situation where a solvable group $G$ has a faithful irreducible character $\chi$ such that $\chi \overline{\chi }$ has exactly two distinct nonprincipal irreducible constituents. We prove that $G$ has derived length bounded above by 8, and provide an example of such a group having derived length 8. In particular, this improves upon a result of Adan-Bante.

Let $p$ be an odd prime, and let $G$ be a nilpotent group, whose commutator subgroup is finite abelian satisfying $G^{\prime }={\mathrm{\text{Syl}}}_p(G)$ and ${\gamma }_3(G)\subseteq {(G^{\prime })}^p$. In this contribution, an upper bound is given on the derived length of the group of units of the group algebra of $G$ over a field of characteristic $p$. Furthermore, we show that this bound is achieved, whenever $G^{\prime }$ is cyclic.

A polycyclic group $G$ is called an $NAF{Q}_n$-group if every normal abelian subgroup of any finite quotient of $G$ is generated by $n$, or fewer, elements and $n$ is the least integer with this property. In this paper, the structure of $NAF{Q}_1$-groups and $NAF{Q}_2$-groups is determined.

A polycyclic group $G$ is called a $NAF{Q}_n$-group ($AF{Q}_n$-group) if every normal abelian subgroup (abelian subgroup) of any finite quotient of $G$ is generated by $n$, or fewer, elements and $n$ is the least integer with this property. In this paper, we describe the structures of $NAF{Q}_3$-groups and $AF{Q}_3$-groups, and bound the number of generators of $AF{Q}_3$-groups and the derived lengths of $NAF{Q}_3$-groups, which is a continuation of [H. G. Liu, F. Zhou and T. Xu, On some polycyclic groups with small Hirsch length, J. Algebra Appl. 16(11) (2017) 17502371–175023710].

In this paper, we study $\pi$-separable groups that every irreducible Isaacs $\pi$-partial character is monomial which we call $M_{\pi }$-group. We show that every $M_{\pi }$-group with solvable $O_{{\pi }^{\prime }}(G)$, is solvable and we get an upper bound for derived length of these groups. Also, we prove that if $N⊴G$ such that $(|N|,|G:N|)$ is ${\pi }^{\prime }$-number, then $N$ is $M_{\pi }$-group.

In this paper, we study solvable groups in which $r_n(F)$ is at most 2. In particular, we investigated groups of odd order and $A_4$-free groups with this property. Exact estimations of the derived length and nilpotent length of such groups are obtained.