arXiv subjects

Alice C. Niemeyer

Publications and source records attributed to Alice C. Niemeyer.

At least 37 records · Page 2Linked to original sources

arXiv2022

The probability of spanning a classical space by two non-degenerate subspaces of complementary dimension

Let $n,n'$ be positive integers and let $V$ be an $(n+n')$-dimensional vector space over a finite field $\mathbb{F}$ equipped with a non-degenerate alternating, hermitian or quadratic form. We estimate the proportion of pairs $(U, U')$, where $U$ is a non-degenerate $n$-subspace and $U'$ is a non-degenerate $n'$-subspace of $V$, such that $U+ U'=V$ (usually such spaces $U$ and $U'$ are not perpendicular). The proportion is shown to be at least $1-c/|\mathbb{F}|$ for some constant $c\leqslant 2$ in the symplectic or unitary cases, and $c<3$ in the orthogonal case.

math.GR↗

arXiv2021

Conjugacy classes and centralisers in wreath products

In analogy to the disjoint cycle decomposition in permutation groups, Ore and Specht define a decomposition of elements of the full monomial group and exploit this to describe conjugacy classes and centralisers of elements in the full monomial group. We generalise their results to wreath products whose base group need not be finite and whose top group acts faithfully on a finite set. We parameterise conjugacy classes and centralisers of elements in such wreath products explicitly. For finite wreath products, our approach yields efficient algorithms for finding conjugating elements, conjugacy classes, and centralisers.

math.GR↗

arXiv2019

A family of hemisystems on the parabolic quadrics

We constuct a family of hemisystems of the parabolic quadric $\mathcal{Q}(2d, q)$, for all ranks $d \ge 2$ and all odd prime powers $q$, that admit $Ω_3(q) \cong \mathrm{PSL}_2(q)$. This yields the first known construction for $d \ge 4$.

math.CO↗

arXiv2019

The icosahedra of edge length 1

Retaining the combinatorial Euclidean structure of a regular icosahedron, namely the 20 equiangular (planar) triangles, the 30 edges of length 1, and the 12 different vertices together with the incidence structure, we investigate variations of the regular icosahedron admitting self-intersections of faces. We determine all rigid equivalence classes of these icosahedra with non-trivial automorphism group and find one curve of flexible icosahedra. Visualisations and explicit data for this paper are available under http://algebra.data.rwth-aachen.de/Icosahedra/visualplusdata.html.

math.GR↗

arXiv2019

Irreducible linear subgroups generated by pairs of matrices with large irreducible submodules

We call an element of a finite general linear group $ \textrm{GL}(d,q) $ \emph{fat} if it leaves invariant, and acts irreducibly on, a subspace of dimension greater than $d/2$. Fatness of an element can be decided efficiently in practice by testing whether its characteristic polynomial has an irreducible factor of degree greater than $d/2$. We show that for groups $G$ with $ \textrm{SL}(d,q) \leq G \leq \textrm{GL}(d,q) $ most pairs of fat elements from $G$ generate irreducible subgroups, namely we prove that the proportion of pairs of fat elements generating a reducible subgroup, in the set of all pairs in $ G \times G $, is less than $q^{-d+1}$. We also prove that the conditional probability to obtain a pair $(g_1,g_2)$ in $G \times G$ which generates a reducible subgroup, given that $g_1, g_2$ are fat elements, is less than $2q^{-d+1}$. Further, we show that any reducible subgroup generated by a pair of fat elements acts irreducibly on a subspace of dimension greater than $ d/2 $, and in the induced action the generating pair corresponds to a pair of fat elements.

math.GR↗

arXiv2018

A Finite Soluble Quotient Algorithm

An algorithm for computing power conjugate presentations for finite soluble quotients of predetermined structure of finitely presented groups is described. Practical aspects of an implementation are discussed.

math.GR↗

arXiv2017

On the second-largest Sylow subgroup of a finite simple group of Lie type

Let $T$ be a finite simple group of Lie type in characteristic $p$, and let $S$ be a Sylow subgroup of $T$ with maximal order. It is well known that $S$ is a Sylow $p$-subgroup except in an explicit list of exceptions, and that $S$ is always `large' in the sense that $|T|^{1/3} < |S| \leqslant |T|^{1/2}$. One might anticipate that, moreover, the Sylow $r$-subgroups of $T$ with $r \neq p$ are usually significantly smaller than $S$. We verify this hypothesis by proving that for every $T$ and every prime divisor $r$ of $|T|$ with $r \neq p$, the order of the Sylow $r$-subgroup of $T$ at most $|T|^{2\lfloor\log_r(4(\ell+1) r)\rfloor/\ell}=|T|^{{\rm O}(\log_r(\ell)/\ell)}$, where $\ell$ is the Lie rank of $T$.

math.GR↗

arXiv2017

Maximal linear groups induced on the Frattini quotient of a $p$-group

Let $p>3$ be a prime. For each maximal subgroup $H\leqslant\mathrm{GL}(d,p)$ with $|H| \geqslant p^{3d+1}$, we construct a $d$-generator finite $p$-group $G$ with the property that $\mathrm{Aut}(G)$ induces $H$ on the Frattini quotient $G/Φ(G)$ and $|G| \leqslant p^{\frac{d^4}{2}}$. A significant feature of this construction is that $|G|$ is very small compared to $|H|$, shedding new light upon a celebrated result of Bryant and Kovács. The groups $G$ that we exhibit have exponent $p$, and of all such groups $G$ with the desired action of $H$ on $G/Φ(G)$, the construction yields groups with smallest nilpotency class, and in most cases, the smallest order.

math.GR↗

arXiv2017

On permutations of order dividing a given integer

We give a detailed analysis of the proportion of elements in the symmetric group on $n$ points whose order divides $m$, for $n$ sufficiently large and $m \ge n$ with $m = O(n)$.

math.GR↗

arXiv2016

The Divisibility Graph of finite groups of Lie Type

The Divisibility Graph of a finite group $G$ has vertex set the set of conjugacy class lengths of non-central elements in $G$ and two vertices are connected by an edge if one divides the other. We determine the connected components of the Divisibility Graph of the finite groups of Lie type in odd characteristic.

math.GR↗

arXiv2016

Primitive prime divisors and the $n$-th cyclotomic polynomial

Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $Φ^*_n(q)$, which is closely related to the cyclotomic polynomial $Φ_n(x)$ and to primitive prime divisors of $q^n-1$. Our definition of $Φ^*_n(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we give an algorithm for determining all pairs $(n,q)$ with $Φ^*_n(q)\le cn^k$. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgroups of finite linear groups.

math.NT↗

arXiv2016

On the Complexity of Multiplication in the Iwahori--Hecke Algebra of the Symmetric Group

We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$.

math.GR↗

arXiv2015

Apportionment methods

Most democratic countries use election methods to transform election results into whole numbers which usually give the number of seats in a legislative body the parties obtained. Which election method does this best can be specified by measuring the error between the allocated result and the ideal proportion. We show how to find an election method which is best suited to a given error function. We also discuss several properties of election methods that have been labelled paradoxa. In particular we explain the highly publicised ``Alabama'' Paradox for the Hare/Hamilton method and show that other popular election methods come with their very own paradoxa.

math.HO↗

arXiv2015

Finding involutions with small support

We show that the proportion of permutations $g$ in $S_n$ or $A_n$ such that $g$ has even order and $g^{|g|/2}$ is an involution with support of cardinality at most $\lceil n^\varepsilon \rceil$ is at least a constant multiple of $\varepsilon$. Using this result, we obtain the same conclusion for elements in a classical group of natural dimension $n$ in odd characteristic that have even order and power up to an involution with $(-1)$-eigenspace of dimension at most $\lceil n^\varepsilon \rceil$ for a linear or unitary group, or $2\lceil \lfloor n/2 \rfloor^\varepsilon \rceil$ for a symplectic or orthogonal group.

math.GR↗

arXiv2015

Identifying long cycles in finite alternating and symmetric groups acting on subsets

Let $H$ be a permutation group on a set $Λ$, which is permutationally isomorphic to a finite alternating or symmetric group $A_n$ or $S_n$ acting on the $k$-element subsets of points from $\{1,\ldots,n\}$, for some arbitrary but fixed $k$. Suppose moreover that no isomorphism with this action is known. We show that key elements of $H$ needed to construct such an isomorphism $φ$, such as those whose image under $φ$ is an $n$-cycle or $(n-1)$-cycle, can be recognised with high probability by the lengths of just four of their cycles in $Λ$.

math.GR↗

arXiv2014

On complexity of multiplication in finite soluble groups

We determine a reasonable upper bound for the complexity of collection from the left to multiply two elements of a finite soluble, or polycyclic, group by restricting attention to certain polycyclic presentations of the group.

math.GR↗

arXiv2014

Elements in finite classical groups whose powers have large 1-Eigenspaces

We estimate the proportion of several classes of elements in finite classical groups which are readily recognised algorithmically, and for which some power has a large fixed point subspace and acts irreducibly on a complement of it. The estimates are used in complexity analyses of new recognition algorithms for finite classical groups in arbitrary characteristic.

math.GR↗

arXiv2013

Fast recognition of alternating groups of unknown degree

We present a constructive recognition algorithm to decide whether a given black-box group is isomorphic to an alternating or a symmetric group without prior knowledge of the degree. This eliminates the major gap in known algorithms, as they require the degree as additional input. Our methods are probabilistic and rely on results about proportions of elements with certain properties in alternating and symmetric groups. These results are of independent interest; for instance, we establish a lower bound for the proportion of involutions with small support.

math.GR↗