Enumerating limit groups

Enumerating Regular Objects associated with Suzuki Groups

We use the Möbius function of the simple Suzuki group Sz(q) to enumerate regular objects such as maps, hypermaps, dessins d'enfants and surface coverings with automorphism groups isomorphic to Sz(q).

Enumerating conjugacy classes of graphical groups over finite fields

Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to their sizes.

Zeta functions of finite groups by enumerating subgroups

For a finite group $G$, we consider the zeta function $ζ_G(s) = \sum_{H} \abs{H}^{-s}$, where $H$ runs over the subgroups of $G$. First we give simple examples of abelian $p$-group $G$ and non-abelian $p$-group $G'$ of order $p^m, \; m \geq 3$ for odd $p$ (resp. $2^m, \; m \geq 4$) for which $ζ_G(s) = ζ_{G'}(s)$. Hence we see there are many non-abelian groups whose zeta functions have symmetry and Euler product, like the case of abelian groups. On the other hand, we show that $ζ_G(s)$ determines the isomorphism class of $G$ within abelian groups, by estimating the number of subgroups of abelia…

Enumerating traceless matrices over compact discrete valuation rings

We enumerate traceless square matrices over finite quotients of compact discrete valuation rings by their image sizes. We express the associated rational generating functions in terms of statistics on symmetric and hyperoctahedral groups, viz. Coxeter groups of types A and B, respectively. These rational functions may also be interpreted as local representation zeta functions associated to the members of an infinite family of finitely generated class-2-nilpotent groups. As a byproduct of our work, we obtain descriptions of the numbers of traceless square matrices over a finite field of fixed r…

Enumerating and identifying semiperfect colorings of symmetrical patterns

If $G$ is the symmetry group of an uncolored pattern then a coloring of the pattern is semiperfect if the associated color group $H$ is a subgroup of $G$ of index 2. We give results on how to identify and enumerate all inequivalent semiperfect colorings of certain patterns. This is achieved by treating a coloring as a partition $\{hJ_iY_i:i\in I,h\in H\}$ of $G$, where $H$ is a subgroup of index 2 in $G$, $J_i\leq H$ for $i\in I$, and $Y=\cup_{i\in I}{Y_i}$ is a complete set of right coset representatives of $H$ in $G$. We also give a one-to-one correspondence between inequivalent semiperfect …

Enumerating finite racks, quandles and kei

A rack of order $n$ is a binary operation $\rack$ on a set $X$ of cardinality $n$, such that right multiplication is an automorphism. More precisely, $(X,\rack)$ is a rack provided that the map $x\mapsto x\rack y$ is a bijection for all $y\in X$, and $(x\rack y)\rack z=(x\rack z)\rack (y\rack z)$ for all $x,y,z\in X$. The paper provides upper and lower bounds of the form $2^{cn^2}$ on the number of isomorphism classes of racks of order $n$. Similar results on the number of isomorphism classes of quandles and kei are obtained. The results of the paper are established by first showing how an arb…

Computational Enumeration of Independent Generating Sets of Finite Symmetric Groups

We developed computer algebra tools for enumerating conjugacy classes of independent subsets and generating sets of symmetric groups up to $n=7$, and carried out an initial analysis of the obtained results.

Enumerating Palindromes and Primitives in Rank Two Free Groups

Let $F= < a,b>$ be a rank two free group. A word $W(a,b)$ in $F$ is {\sl primitive} if it, along with another group element, generates the group. It is a {\sl palindrome} (with respect to $a$ and $b$) if it reads the same forwards and backwards. It is known that in a rank two free group any primitive element is conjugate either to a palindrome or to the product of two palindromes, but known iteration schemes for all primitive words give only a representative for the conjugacy class. Here we derive a new iteration scheme that gives either the unique palindrome in the conjugacy class or expresse…

On representation zeta functions of groups and a conjecture of Larsen and Lubotzky

We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to certain p-adic analytic pro-p groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding the abscissa of convergence of arithmetic groups of type A_2 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.

Local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms

We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.

Adventures in Invariant Theory

We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualised via two case studies arising from our recent work: entanglement measures, for characterising the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.

The rationality of Sol manifolds

Let $Γ$ be the fundamental group of a manifold modeled on three dimensional Sol geometry. We prove that $Γ$ has a finite index subgroup $G$ which has a rational growth series with respect to a natural generating set. We do this by enumerating $G$ by a regular language. However, in contrast to most earlier proofs of this sort our regular language is not a language of words in the generating set, but rather reflects a different geometric structure in $G$.

居间关系与$D$-集的Omega-范畴极限

Omega-categorical limits of betweenness relations and $D$-sets

本文研究两类寡形Jordan置换群构造，解决了相关开放问题，证明$D$-关系极限可齐性化，明确其NIP性质、轨道增长速率等特征，并验证其自同构群为对称群中的极大闭子群。