arXiv subjects
Publications and source records attributed to Felix Leinen.
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.