In this paper we introduce the concept of α-commutator which its definition is based on generalized conjugate classes. With this notion, α-nilpotent groups, α-solvable groups, nilpotency and solvability of groups related to the automorphism are defined. N(G) and S(G) are the set of all nilpotency classes and the set of all solvability classes for the group G with respect to different automorphi...

let $g$ be a $p$-group of order $p^n$ and $phi$=$phi(g)$ be the frattini subgroup of $g$. it is shown that the nilpotency class of $autf(g)$, the group of all automorphisms of $g$ centralizing $g/ fr(g)$, takes the maximum value $n-2$ if and only if $g$ is of maximal class. we also determine the nilpotency class of $autf(g)$ when $g$ is a finite abelian $p$-group.

We investigate the computational properties of cellular automata on countable (equivalently, zero entropy) sofic shifts with an emphasis on nilpotency, periodicity, and asymptotic behavior. As a tool for proving decidability results, we prove the Starfleet Lemma, which is of independent interest. We present computational results including the decidability of nilpotency and periodicity, the unde...

Four infinite families of 2-groups are presented, all of whose members possess an outer automorphism that preserves conjugacy classes. The groups in these families are central extensions of their predecessors by a cyclic group of order 2. In particular, for each integer r > 1, there is precisely one 2-group of nilpotency class r in each of the four families. All other known families of 2-groups...

By T. Kepka and M. Niemenmaa if the inner mapping group of a finite loop Q is abelian, then the loop Q is centrally nilpotent. For a long time there was no example of a nilpotency degree greater than two. In the nineties T. Kepka raised the following problem: whether every finite loop with abelian inner mapping group is centrally nilpotent of class at most two. For many years the prevailing opi...

In 2004, Csörgő constructed a loop of nilpotency class three with abelian group of inner mappings. As of now, no other examples are known. We construct many such loops from groups of nilpotency class two by replacing the product xy with xyh in certain positions, where h is a central involution. The location of the replacements is ultimately governed by a symmetric trilinear alternating form. c ...

Let $G$ be a $p$-group of order $p^n$ and $Phi$=$Phi(G)$ be the Frattini subgroup of $G$. It is shown that the nilpotency class of $Autf(G)$, the group of all automorphisms of $G$ centralizing $G/ Fr(G)$, takes the maximum value $n-2$ if and only if $G$ is of maximal class. We also determine the nilpotency class of $Autf(G)$ when $G$ is a finite abelian $p$-group.

In this paper, we continue a study of simplicial commutative algebras with finite André-Quillen homology, that was begun in [19]. Here we restrict our focus to simplicial algebras having characteristic 2. Our aim is to find a generalization of the main theorem in [19]. In particular, we replace the finiteness condition on homotopy with a weaker condition expressed in terms of nilpotency for the...

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant we compare with classical cocategory, but also with the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to character...

Let K be a field of positive characteristic p and KG the group algebra of a group G. It is known that if KG is Lie nilpotent then its upper (or lower) Lie nilpotency index is at most |G| + 1, where |G| is the order of the commutator subgroup. The class of groups G for which these indices are maximal or almost maximal have already been determined. Here we determine G for which upper (or lower) L...