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Martin W. Liebeck

Publications and source records attributed to Martin W. Liebeck.

34 records · Page 2Linked to original sources
arXiv2018

The length and depth of algebraic groups

Let $G$ be a connected algebraic group. An unrefinable chain of $G$ is a chain of subgroups $G = G_0 > G_1 > \cdots > G_t = 1$, where each $G_i$ is a maximal connected subgroup of $G_{i-1}$. We introduce the notion of the length (respectively, depth) of $G$, defined as the maximal (respectively, minimal) length of such a chain. Working over an algebraically closed field, we calculate the length of a connected group $G$ in terms of the dimension of its unipotent radical $R_u(G)$ and the dimension of a Borel subgroup $B$ of the reductive quotient $G/R_u(G)$. In particular, a simple algebraic group of rank $r$ has length $\dim B + r$, which gives a natural extension of a theorem of Solomon and Turull on finite quasisimple groups of Lie type. We then deduce that the length of any connected algebraic group $G$ exceeds $\frac{1}{2} \dim G$. We also study the depth of simple algebraic groups. In characteristic zero, we show that the depth of such a group is at most $6$ (this bound is sharp). In the positive characteristic setting, we calculate the exact depth of each exceptional algebraic group and we prove that the depth of a classical group (over a fixed algebraically closed field of positive characteristic) tends to infinity with the rank of the group. Finally we study the chain difference of an algebraic group, which is the difference between its length and its depth. In particular we prove that, for any connected algebraic group $G$, the dimension of $G/R(G)$ is bounded above in terms of the chain difference of $G$.

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arXiv2018

Base sizes of primitive groups: bounds with explicit constants

We show that the minimal base size $b(G)$ of a finite primitive permutation group $G$ of degree $n$ is at most $2 (\log |G|/\log n) + 24$. This bound is asymptotically best possible since there exists a sequence of primitive permutation groups $G$ of degrees $n$ such that $b(G) = \lfloor 2 (\log |G|/\log n) \rceil - 2$ and $b(G)$ is unbounded. As a corollary we show that a primitive permutation group of degree $n$ that does not contain the alternating group $\mathrm{Alt}(n)$ has a base of size at most $\max\{\sqrt{n} , \ 25\}$.

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arXiv2018

Bases of quasisimple linear groups

Let $V$ be a vector space of dimension $d$ over $F_q$, a finite field of $q$ elements, and let $G \le GL(V) \cong GL_d(q)$ be a linear group. A base of $G$ is a set of vectors whose pointwise stabiliser in $G$ is trivial. We prove that if $G$ is a quasisimple group (i.e. $G$ is perfect and $G/Z(G)$ is simple) acting irreducibly on $V$, then excluding two natural families, $G$ has a base of size at most 6. The two families consist of alternating groups ${\rm Alt}_m$ acting on the natural module of dimension $d = m-1$ or $m-2$, and classical groups with natural module of dimension $d$ over subfields of $F_q$.

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arXiv2017

The depth of a finite simple group

We introduce the notion of the depth of a finite group $G$, defined as the minimal length of an unrefinable chain of subgroups from $G$ to the trivial subgroup. In this paper we investigate the depth of (non-abelian) finite simple groups. We determine the simple groups of minimal depth, and show, somewhat surprisingly, that alternating groups have bounded depth. We also establish general upper bounds on the depth of simple groups of Lie type, and study the relation between the depth and the much studied notion of the length of simple groups. The proofs of our main theorems depend (among other tools) on a deep number-theoretic result, namely, Helfgott's recent solution of the ternary Goldbach conjecture.

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arXiv2017

Permutation representations of nonsplit extensions involving alternating groups

L. Babai has shown that a faithful permutation representation of a nonsplit extension of a group by an alternating group $A_k$ must have degree at least $k^2(\frac{1}{2}-o(1))$, and has asked how sharp this lower bound is. We prove that Babai's bound is sharp (up to a constant factor), by showing that there are such nonsplit extensions that have faithful permutation representations of degree $\frac{3}{2}k(k-1)$. We also reprove Babai's quadratic lower bound with the constant $\frac{1}{2}$ improved to 1 (by completely different methods).

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arXiv2017

Algorithms determining finite simple images of finitely presented groups

We address the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for families of simple groups of Lie type of bounded rank, we obtain positive results. For example, given any fixed untwisted Lie type $X$ there is an algorithm that determines whether or not an arbitrary finitely presented group has infinitely many simple images isomorphic to $X(q)$ for some $q$, and if there are finitely many, the algorithm determines them.

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arXiv2017

Character bounds for finite groups of Lie type

We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form $|χ(g)| \le χ(1)^{α_g}$, and give rise to a variety of applications, for example to covering numbers and mixing times of random walks on such groups. In particular we deduce that, if $G$ is a classical group in dimension $n$, then, under some conditions on $G$ and $g \in G$, the mixing time of the random walk on $G$ with the conjugacy class of $g$ as a generating set is (up to a small multiplicative constant) $n/s$, where $s$ is the support of $g$.

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arXiv2016

Generation of second maximal subgroups and the existence of special primes

Let $G$ be a finite almost simple group. It is well known that $G$ can be generated by 3 elements, and in previous work we showed that 6 generators suffice for all maximal subgroups of $G$. In this paper we consider subgroups at the next level of the subgroup lattice - the so-called second maximal subgroups. We prove that with the possible exception of some families of rank 1 groups of Lie type, the number of generators of every second maximal subgroup of $G$ is bounded by an absolute constant. We also show that such a bound holds without any exceptions if and only if there are only finitely many primes $r$ for which there is a prime power $q$ such that $(q^r-1)/(q-1)$ is prime. The latter statement is a formidable open problem in Number Theory. Applications to random generation and polynomial growth are also given.

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arXiv2016

Chiral polyhedra and finite simple groups

We prove that every finite non-abelian simple group acts as the automorphism group of a chiral polyhedron, apart from the groups $PSL_2(q)$, $PSL_3(q)$, $PSU_3(q)$ and $A_7$.

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arXiv2014

The classification of (3/2)-transitive permutation groups and (1/2)-transitive linear groups

A linear group G on a finite vector space V, (that is, a subgroup of GL(V)) is called (1/2)-transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the (1/2)-transitive linear groups. As a consequence we complete the determination of the finite (3/2)-transitive permutation groups -- the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the finite (k+1/2)-transitive permutation groups for integers k > 1.

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arXiv2014

Recognition of finite exceptional groups of Lie type

Let $q$ be a prime power and let $G$ be an absolutely irreducible subgroup of $GL_d(F)$, where $F$ is a finite field of the same characteristic as $\F_q$, the field of $q$ elements. Assume that $G \cong G(q)$, a quasisimple group of exceptional Lie type over $\F_q$ which is neither a Suzuki nor a Ree group. We present a Las Vegas algorithm that constructs an isomorphism from $G$ to the standard copy of $G(q)$. If $G \not\cong {}^3 D_4(q)$ with $q$ even, then the algorithm runs in polynomial time, subject to the existence of a discrete log oracle.

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arXiv2014

Arithmetic results on orbits of linear groups

Let $p$ be a prime and $G$ a subgroup of $GL_d(p)$. We define $G$ to be $p$-exceptional if it has order divisible by $p$, but all its orbits on vectors have size coprime to $p$. We obtain a classification of $p$-exceptional linear groups. This has consequences for a well known conjecture in representation theory, and also for a longstanding question concerning 1/2-transitive linear groups (i.e. those having all orbits on nonzero vectors of equal length), classifying those of order divisible by $p$.

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arXiv2011

The classification of almost simple $\tfrac{3}{2}$-transitive groups

A finite transitive permutation group is said to be 3/2-transitive if all the nontrivial orbits of a point stabilizer have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less obvious ones. We prove that 3/2-transitive groups are either affine or almost simple, and classify the latter. One of the main steps in the proof is an arithmetic result on the subdegrees of groups of Lie type in characteristic $p$: with some explicitly listed exceptions, every primitive action of such a group is either 2-transitive, or has a subdegree divisible by $p$.

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