arXiv subjects
Publications and source records attributed to Orazio Puglisi.
Let G be a finite solvable group, and let h(G) denote its Fitting height, namely the length of a shortest normal series in G with nilpotent factors. We show, that any law in G has length at least h(G). This result is then used to improve a previously given bound on the nonsolvable length of finite nonsolvable groups.
Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of non-abelian simple groups. The minimum number of nonsolvable factors, attained on all possible such series in $G$, is called the \emph{nonsolvable length} $λ(G)$ of $G$. In the present paper, we prove a theorem about permutation representations of groups of fixed nonsolvable length. As a consequence, we show that in a finite group of nonsolvable length at least $n$, no non-trivial word of length at most $n$ (in any number of variables) can be a law. This result is then used to give a bound on $λ(G)$ in terms of the length of the shortest law of $G$, thus confirming a conjecture of Larsen. Moreover our Theorem C can be used to give a positive answer, in the case $p=2$, to a problem raised by Khukhro and Shumyatsky, concerning the non-$p$-solvable length of finite groups.
Every finite solvable group $G$ has a normal series with nilpotent factors. The smallest possible number of factors in such a series is called the Fitting height $h(G)$. In the present paper, we derive an upper bound for $h(G)$ in terms of the exponent of $G$. Our bound constitutes a considerable improvement of an earlier bound obtained by Shalev.
Every finite group $G$ has a normal series each of whose factors is either a solvable group or a direct product of nonabelian simple groups. The minimum number of nonsolvable factors attained on all possible such series is called the nonsolvable length of the group and denoted by $λ(G)$. For every integer $n$, we define a particular class of groups of nonsolvable length $n$, called \emph{$n$-rarefied}, and we show that every finite group of nonsolvable length $n$ contains an $n$-rarefied subgroup. As applications of this result, we improve the known upper bounds on $λ(G)$ and determine the maximum possible nonsolvable length for permutation groups and linear groups of fixed degree resp. dimension.
Following Plotkin we say that the automorphism $x$ of the group $G$ is a nil-automorphism if, for every $g\in G$, there exists $n=n(g)$ such that $[g,_n x]=1$. If the integer $n$ can be chosen independently of $g$, then $x$ is said to be unipotent. Nil and unipotent automorphisms can be regarded as a natural extension of the concept of Engel element, since a nil-automorphism $x$ is just a left-Engel element in $G < x >$. In this paper we consider nil-automorphisms of groups with residual properties namely locally-graded groups, residually-finite groups and profinite groups. The first result we prove says that a finite group of nil-automorphisms of a locally graded group, must be nilpotent. Next we turn our attention to groups of unipotent automorphisms of residually-finite and profinite groups. We show that such groups are locally-nilpotent and, as a by-product, we obtain an alternative proof of a well known theorem of Wilson about n-Engel residually-finite groups.
We study the Hirsch-Plotkin radical of stability groups of (general) subspace series of infinite dimensional vector spaces. We show that in countable dimension and some other cases, the HP-radical of the stability group coincides with the set of all space automorphisms that fix a finite sub-series; this implies that the HP radical is a Fitting group. Conversely, we prove that every countable Fitting group, which is either torsion-free or a p-group may be represented as a subgroup of the HP radical of a series stabilizer.