A finite group $G$ is called {\it $p^i$-central of height $k$} if every element of order $p^i$ of $G$ is contained in the $k^{th}$-term $ζ_k(G)$ of the ascending central series of $G$. If $p$ is odd such a group has to be $p$-nilpotent (Thm. A). Finite $p$-central $p$-groups of height $p-2$ can be seen as the dual analogue of finite potent $p$-groups, i.e., for such a finite $p$-group $P$ the group $P/Ω_1(P)$ is also $p$-central of height $p-2$ (Thm. B). In such a group $P$ the index of $P^p$ is less or equal than the order of the subgroup $Ω_1(P)$ (Thm. C). If the Sylow $p$-subgroup $P$ of a finite group $G$ is $p$-central of height $p-1$, $p$ odd, and $N_G(P)$ is $p$-nilpotent, then $G$ is also $p$-nilpotent (Thm. D). Moreover, if $G$ is a $p$-soluble finite group, $p$ odd, and $P\in \text{Syl}_p(G)$ is $p$-central of height $p-2$, then $N_G(P)$ controls $p$-fusion in $G$ (Thm. E). It is well-known that the last two properties hold for Swan groups.
We prove two conjectures of E. Khukhro and P. Shumyatsky concerning the Fitting height and insoluble length of finite groups. As a by-product of our methods, we also prove a generalization of a result of Flavell, which itself generalizes Wielandt's Zipper Lemma and provides a characterization of subgroups contained in a unique maximal subgroup. We also derive a number of consequences of our theorems, including some applications to the set of odd order elements of a finite group inverted by an involutory automorphism.
The $\mathcal{R}$-height of a semigroup $S$ is the height of the poset of $\mathcal{R}$-classes of $S,$ i.e. the supremum of the lengths of chains of $\mathcal{R}$-classes. Given a semigroup $S$ with finite $\mathcal{R}$-height, we establish bounds on the $\mathcal{R}$-height of bi-ideals, one-sided ideals and two-sided ideals; in particular, these substructures inherit the property of having finite $\mathcal{R}$-height. We then investigate whether these bounds can be attained.
Motivated by classic theorems of Thompson and Berger on the Fitting height of finite groups with a fixed-point-free automorphism of coprime order, we conjecture that, for every non-zero polynomial $f(x) = a_0 + a_1 x + \cdots + a_d x^d \in \mathbb{Z}[x] $, there is an integer $k > 0$ with the following property. Let $G$ be a finite (solvable) group with a fixed-point-free automorphism $α$ satisfying $\gcd(|G|,k)= 1$ and $$\{ g^{a_0} \cdot α(g)^{a_1} \cdot α^2(g)^{a_2} \cdots α^d(g)^{a_d} | g \in G \} = \{1\}.$$ Then the Fitting height of $G$ is at most the number of irreducible factors of $f(x…
The paper is concerned with Kropholler's conjecture on splitting a finitely generated group over a codimension-1 subgroup. For a subgroup H of a group G, we define the notion of "finite splitting height" which generalises the finite-height property. By considering the dual CAT(0) cube complex associated to a codimension-1 subgroup H in G, we show that the Kropholler-Roller conjecture holds when H has finite splitting height in G. Examples of subgroups of finite height are stable subgroups or more generally strongly quasiconvex subgroups. Examples of subgroups of finite splitting height include…
Let $f(x)$ be a non-zero polynomial with integer coefficients. An automorphism $φ$ of a group $G$ is said to satisfy the elementary abelian identity $f(x)$ if the linear transformation induced by $φ$ on every characteristic elementary abelian section $S$ of $G$ is annihilated by $f(x)$. We prove that if a finite (soluble) group $G$ admits a fixed-point-free automorphism $φ$ satisfying an elementary abelian identity $f(x)$, where $f(x)$ is a primitive polynomial, then the Fitting height of $G$ is bounded in terms of $\operatorname{deg}(f(x))$. We also prove that if $f(x)$ is any non-zero polyno…
In this paper, we prove a combination theorem for a relatively acylindrical graph of relatively hyperbolic groups (Theorem 1.1). Here, we are extending the technique of [Tom21] and constructing Bowditch boundary of the fundamental group of graph of groups. Suppose G(Y) is a graph of relatively hyperbolic groups such that edge groups are relatively quasi-convex in adjacent vertex groups. Also, assume that the fundamental group of G(Y) is relatively hyperbolic. Then we show that the edge groups of G(Y) have finite relative height (Definition 1.5) if and only if they are relatively quasi-convex (…
The nonsoluble length $λ(G)$ of a finite group $G$ is defined as the number of nonsoluble factors in a shortest normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. The generalized Fitting height of a finite group $G$ is the least number $h=h^*(G)$ such that $F^*_h(G)=G$, where $F^*_1(G)=F^*(G)$ is the generalized Fitting subgroup, and $F^*_{i+1}(G)$ is the inverse image of $F^*(G/F^*_{i}(G))$. It is proved that if a finite group $G=AB$ is factorized by two subgroups of coprime orders, then the nonsoluble length of $G$ is bounded in terms of…
Eaton and Moretó proposed an extension of Brauer's famous height zero conjecture on blocks of finite groups to the case of non-abelian defect groups, which predicts the smallest non-zero height in such blocks in terms of local data. We show that their conjecture holds for principal blocks of quasi-simple groups, for all blocks of finite reductive groups in their defining characteristic, as well as for all covering groups of symmetric and alternating groups. For the proof, we determine the minimal non-trivial character degrees of Sylow $p$-subgroups of finite reductive groups in characteristic~…
In this paper we are concerned with finite soluble groups $G$ admitting a factorisation $G=AB$, with $A$ and $B$ proper subgroups having coprime order. We are interested in bounding the Fitting height of $G$ in terms of some group-invariants of $A$ and $B$: including the Fitting heights and the derived lengths.
The generalized Fitting height $h^*(G)$ of a finite group $G$ is the least number $h$ such that $\mathrm{F}_h^* (G) = G$, where $\mathrm{F}_{(0)}^* (G) = 1$, and $\mathrm{F}_{(i+1)}^*(G)$ is the inverse image of the generalized Fitting subgroup $\mathrm{F}^*(G/\mathrm{F}^*_{(i)} (G))$. Let $p$ be a prime, $1=G_0\leq G_1\leq\dots\leq G_{2h+1}=G$ be the shortest normal series in which for $i$ odd the factor $G_{i+1}/G_i$ is $p$-soluble (possibly trivial), and for $i$ even the factor $G_{i+1}/G_i$ is a (non-empty) direct product of nonabelian simple groups. Then $h=λ_p(G)$ is called the non-$p$-s…
The action of a finite reflection group (type A) on its set of roots is understood as a permutation representation or group action. We show that this representation is an induced representation from a certain kind of parabolic subgroup. Furthermore, we use this representation to define a statistic (derived from the length function) on the set of roots. A possible application to Costas Arrays is hinted at in a proposition.
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.
We show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height $\hat{h}(F)$ of a finite set of matrices $F$ in $GL_{n}(\bar{\Bbb{Q}})$ which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribution of Galois orbits of small points, that $\hat{h}(F)>ε$ as soon as $F$ generates a non-virtually solvable subgroup of $SL_{n}(\bar{\Bbb{Q}}),$ where $ε=ε(n)>0$ is an absolute constant.
We propose a projective version of the celebrated Brauer's Height Zero Conjecture on characters of finite groups and prove it, among other cases, for $p$-solvable groups as well as for (some) quasi-simple groups.
We show that Brauer's height zero conjecture holds for blocks of finite quasi-simple groups. This result is used in Navarro-Späth's reduction of this conjecture for general groups to the inductive Alperin-McKay condition for simple groups.
This paper has two main results. Firstly, we complete the parametrisation of all p-blocks of finite quasi-simple groups by finding the so-called quasi-isolated blocks of exceptional groups of Lie type for bad primes. This relies on the explicit decomposition of Lusztig induction from suitable Levi subgroups. Our second major result is the proof of one direction of Brauer's long-standing height zero conjecture on blocks of finite groups, using the reduction by Berger and Knörr to the quasi-simple situation. We also use our result on blocks to verify a conjecture of Malle and Navarro on nilpoten…
In [Bl1], it is proved that a subgroup of $PL_{+}(I)$ has a finite height if and only if it is solvable. We prove the "only if" part for any subgroup of Homeo$_{+}(I)$, and present a construction which indicates a plethora of examples of solvable groups with infinite height.
Let $G$ be a permutation group on a set $Ω$ of size $t$. We say that $Λ\subseteqΩ$ is an independent set if its pointwise stabilizer is not equal to the pointwise stabilizer of any proper subset of $Λ$. We define the height of $G$ to be the maximum size of an independent set, and we denote this quantity $\mathrm{H}(G)$. In this paper we study $\mathrm{H}(G)$ for the case when $G$ is primitive. Our main result asserts that either $\mathrm{H}(G)< 9\log t$, or else $G$ is in a particular well-studied family (the "primitive large--base groups"). An immediate corollary of this result is a character…
E. Breuillard showed that finite subsets $F$ of matrices in $GL_d(\overline{\mathbb{Q}})$ generating non-virtually solvable groups have normalized height $\widehat{h}(F) \ge ε_d$, for some positive $ε_d >0$. The normalized height $\widehat{h}(F)$ is a measure of the arithmetic size of $F$ and this result can be thought of as a non-abelian analog of Lehmer's Mahler measure problem. We give a new shorter proof of this result. Our key idea relies on the existence of particular word maps in compact Lie groups (known as almost laws) whose image lies close to the identity element.
In this paper, we study strongly quasiconvex subgroups in a finitely generated $3$--manifold group $π_1(M)$. We prove that if $M$ is a compact, orientable $3$--manifold that does not have a summand supporting the Sol geometry in its sphere-disc decomposition then a finitely generated subgroup $H \le π_1(M)$ has finite height if and only if $H$ is strongly quasiconvex. On the other hand, if $M$ has a summand supporting the Sol geometry in its sphere-disc decomposition then $π_1(M)$ contains finitely generated, finite height subgroups which are not strongly quasiconvex. We also characterize stro…
The connective constant $μ(G)$ of an infinite transitive graph $G$ is the exponential growth rate of the number of self-avoiding walks from a given origin. In earlier work of Grimmett and Li, a locality theorem was proved for connective constants, namely, that the connective constants of two graphs are close in value whenever the graphs agree on a large ball around the origin. A condition of the theorem was that the graphs support so-called 'unimodular graph height functions'. When the graphs are Cayley graphs of infinite, finitely generated groups, there is a special type of unimodular graph …
