arXiv subjects

Simon Guest

Publications and source records attributed to Simon Guest.

14 recordsLinked to original sources

arXiv2025

Constructing flexible polyhedra by twinning

Polyhedra are generically rigid, but can be made to flex under certain symmetry conditions. We generalise Raoul Bricard's 1897 method for making flexible octahedra to construct an infinite family of flexible polyhedra with self-intersections. Removing an edge from any of these models gives a crinkle, and these can be used to create flexible polyhedra without self-interesection. We show this in a particular example, giving a flexible embedded polyhedron with a large range of motion. We also discuss a novel crinkle.

math.MG↗

arXiv2014

Finite primitive groups and regular orbits of group elements

We prove that if $G$ is a finite primitive permutation group and if $g$ is an element of $G$, then either $g$ has a cycle of length equal to its order, or for some $r$, $m$ and $k$, the group $G \leq \mathrm{Sym}(m) \textrm{wr} \mathrm{Sym}(r)$ preserves the product structure of $r$ direct copies of the natural action of $\mathrm{Sym}(m)$ on $k$-sets. This gives an answer to a question of Siemons and Zalesski and a solution to a conjecture of Giudici, Praeger and the second author.

math.GR↗

arXiv2013

Finite primitive permutation groups containing a permutation having at most four cycles

We classify the finite primitive groups containing a permutation with at most four cycles (including fixed points) in its disjoint cycle representation.

math.GR↗

arXiv2013

Affine transformations of finite vector spaces with large orders or few cycles

Let V be a d-dimensional vector space over a field of prime order p. We classify the affine transformations of V of order at least p^d/4, and apply this classification to determine the finite primitive permutation groups of affine type, and of degree n, that contain a permutation of order at least n/4. Using this result we obtain a classification of finite primitive permutation groups of affine type containing a permutation with at most four cycles.

math.GR↗

arXiv2013

A remark on the permutation representations afforded by the embeddings of $O_{2m}^\pm(2^f)$ in $Sp_{2m}(2^f)$

We show that the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on its natural module is isomorphic to the permutation module over $\mathbb{C}$ afforded by the action of $Sp_{2m}(2^f)$ on the union of the right cosets of $O_{2m}^+(2^f)$ and $O_{2m}^-(2^f)$.

math.GR↗

arXiv2013

Criteria for solvable radical membership via p-elements

Guralnick, Kunyavskii, Plotkin and Shalev have shown that the solvable radical of a finite group $G$ can be characterized as the set of all $x\in G$ such that $ $ is solvable for all $y\in G$. We prove two generalizations of this result. Firstly, it is enough to check the solvability of $ $ for every $p$-element $y\in G$ for every odd prime $p$. Secondly, if $x$ has odd order, then it is enough to check the solvability of $ $ for every 2-element $y\in G$.

math.GR↗

arXiv2013

The exact number of r-regular elements in finite exceptional groups

We calculate the precise number of r-regular elements in the finite exceptional groups. As a corollary we find that the proportion of r-regular elements is at least 3577/18432 and for all ε>0, there are infinitely finite simple exceptional groups such that the proportion of r-regular elements is less than $3577/18432+ε$ for some prime r.

math.GR↗

arXiv2013

On the maximum orders of elements of finite almost simple groups and primitive permutation groups

We determine upper bounds for the maximum order of an element of a finite almost simple group with socle T in terms of the minimum index m(T) of a maximal subgroup of T: for T not an alternating group we prove that, with finitely many exceptions, the maximum element order is at most m(T). Moreover, apart from an explicit list of groups, the bound can be reduced to m(T)/4. These results are applied to determine all primitive permutation groups on a set of size n that contain permutations of order greater than or equal to n/4.

math.GR↗

arXiv2012

Proportions of elements with given 2-part order in finite classical groups of odd characteristic

For an element $g$ in a group $X$, we say that $g$ has 2-part order $2^{a}$ if $2^{a}$ is the largest power of 2 dividing the order of $g$. We prove lower bounds on the proportion of elements in finite classical groups in odd characteristic that have certain 2-part orders. In particular, we show that the proportion of odd order elements in the symplectic and orthogonal groups is at least $C/\ell^{3/4}$, where $\ell$ is the Lie rank, and $C$ is an explicit constant. We also prove positive constant lower bounds for the proportion of elements of certain 2-part orders independent of the Lie rank. Furthermore, we describe how these results can be used to analyze part of Yalçinkaya's Black Box recognition algorithm for finite classical groups in odd characteristic.

math.GR↗

arXiv2012

Proportions of r-regular elements in finite classical group

For a prime $r$, we obtain lower bounds on the proportion of $r$-regular elements in classical groups and show that these lower bounds are the best possible lower bounds that do not depend on the order of the defining field. Along the way, we also provide new upper bounds and answer some open questions of the first author, Pálfy and Saxl.

math.GR↗

arXiv2010

Further solvable analogues of the Baer-Suzuki theorem and generation of nonsolvable groups

Let $G$ be an almost simple group. We prove that if $x \in G$ has prime order $p \ge 5$, then there exists an involution $y$ such that $ $ is not solvable. Also, if $x$ is an involution then there exist three conjugates of $x$ that generate a nonsolvable group, unless $x$ belongs to a short list of exceptions, which are described explicitly. We also prove that if $x$ has order $6$ or $9$, then there exists two conjugates that generate a nonsolvable group.

math.GR↗

arXiv2009

Characterizations of the Solvable Radical

We prove that there exists a constant $k$ with the property: if $\calC$ is a conjugacy class of a finite group $G$ such that every $k$ elements of $\calC$\ generate a solvable subgroup then $\calC$ generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take $k=4$. We also present proofs that do not use the Classification theorem. The most direct proof gives a value of $k=10$. By lengthening one of our arguments slightly, we obtain a value of $k=7$.

math.GR↗

arXiv2009

A solvable version of the Baer--Suzuki Theorem

Suppose that G is a finite group and x in G has prime order p > 3. Then x is contained in the solvable radical of G if (and only if) is solvable for all g in G. If G is an almost simple group and x in G has prime order p > 3 then this implies that there exists g in G such that is not solvable. In fact, this is also true when p=3 with very few exceptions, which are described explicitly.

math.GR↗

arXiv2008

When is a symmetric pin-jointed framework isostatic?

Maxwell's rule from 1864 gives a necessary condition for a framework to be isostatic in 2D or in 3D. Given a framework with point group symmetry, group representation theory is exploited to provide further necessary conditions. This paper shows how, for an isostatic framework, these conditions imply very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the framework. In particular, it turns out that an isostatic framework in 2D can belong to one of only six point groups. Some conjectures and initial results are presented that would give sufficient conditions (in both 2D and 3D) for a framework that is realized generically for a given symmetry group to be an isostatic framework.

math.MG↗