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Cindy Tsang

Publications and source records attributed to Cindy Tsang.

At least 19 recordsLinked to original sources
arXiv2026

A probabilistic approach to the Yang--Baxter equation and skew braces

We investigate finite non-degenerate set-theoretic solutions to the Yang--Baxter equation and skew braces using a probabilistic approach. We introduce four probabilities that measure how far a solution is from being a flip, but in different ways. Our main results state that for solutions arising from skew braces, these probabilities exhibit a rigid behaviour --- apart from a finite list of exceptional values (which we show to occur by means of explicit examples), they admit an upper bound that is slightly above $\frac{1}{2}$. In the skew brace setting, these probabilities measure how far the underlying skew brace is from being a trivial brace (in two different ways: one via the annihilator and the other via the socle), a trivial skew brace, and an almost trivial skew brace. We also introduce a probability that is related to the indecomposable components of a solution, and a probability that measures how close an arbitrary bijective non-degenerate map is to being a solution. In contrast to our main results, these two probabilities do not exhibit a discrete behaviour near $1$ --- they can be made arbitrarily close to $1$.

math.GR↗
arXiv2026

Skew braces with no proper left ideals

A skew brace $A = (A,\cdot,\circ)$ is said to be \textit{left-simple} if $A\neq1$ and it has no left ideal other than $1$ and $A$. The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result of Stefanello and Trappeniers implies that finite left-simple skew braces correspond precisely to minimal Hopf--Galois structures on finite Galois extensions of fields.

math.GR↗
arXiv2025

Hopf--Galois structures of cyclic type on parallel extensions of prime power degree

Let $L/K$ be any finite separable extension with normal closure $\widetilde{L}/K$. An extension $L'/K$ is said to be $\textit{parallel to $L/K$}$ if $L'$ is an intermediate field of $\widetilde{L}/K$ with $[L':K]=[L:K]$. We study the following question -- Given that $L/K$ admits a Hopf--Galois structure of type $N$, does it imply that every extension parallel to $L/K$ also admits a Hopf--Galois structure of type $N$? We completely solve this problem when the degree $[L:K]$ is a prime power and the type $N$ is cyclic. Our approach is group-theoretic and uses the work of Greither--Pareigis and Byott.

math.GR↗
arXiv2025

On cardinalities whose arithmetical properties determine the structure of solutions of the Yang--Baxter equation

The aim of this paper is to provide purely arithmetical characterisations of those natural numbers $n$ for which every non-degenerate set-theoretic solution of cardinality $n$ of the Yang--Baxter equation arising from a skew brace (sb-solution for short) satisfies some relevant properties, such as being a flip or being involutive. For example, it turns out that every sb-solution of cardinality $n$ has finite multipermutation level if and only if its prime factorisation $n= p_1^{\alpha_1} \ldots p_t^{\alpha_t}$ is cube-free, namely $\alpha_i\leq 2$ for every $i$, and $p_i$ does not divide $p_j^{\alpha_j}-1$ for $i\neq j$. Two novel constructions of skew braces will play a central role in our proofs. We shall also introduce the notion of supersoluble solution and show how this concept is related to that of supersoluble skew brace. In doing so, we have spotted an irreparable mistake in the proof of Theorem C [Ballester-Bolinches et al., Adv. Math. 455 (2024)], which characterizes soluble solutions in terms of soluble skew braces.

math.GR↗
arXiv2025

On $n$-isoclinism of skew braces

The purpose of this paper is to explore possible definitions of $n$-isoclinism for skew braces. We also introduce the notions of verbal sub-skew braces and marginal left ideals.

math.GR↗
arXiv2025

Analogs of the lower and upper central series in skew braces: a survey

A skew brace is a ring-like and group-like algebraic structure that was introduced in the study of set-theoretic solutions to the Yang-Baxter equation. In this survey paper, we shall consider the left series, right series, socle series, and annihilator series of skew braces. They may be regarded as analogs of the lower and upper central series of groups. Other than some well-known facts regarding these series, we shall prove several new results about the relationship among their terms. We shall also consider the lower central series of skew braces that was defined by Bonatto and Jedli\v{c}ka. As we shall explain, it seems to be the ``correct" analog of the lower central series for skew braces. Concerning this, we shall also discuss the notion of the lower central series of ideals that is due to Ballester-Bolinches et al.

math.GR↗
arXiv2024

The multiple holomorph of centerless groups

Let $G$ be a group. The holomorph $\mathrm{Hol}(G)$ may be defined as the normalizer of the subgroup of either left or right translations in the group of all permutations of $G$. The multiple holomorph $\mathrm{NHol}(G)$ is in turn defined as the normalizer of the holomorph. Their quotient $T(G) = \mathrm{NHol}(G)/\mathrm{Hol}(G)$ has been computed for various families of groups $G$. In this paper, we consider the case when $G$ is centerless, and we show that $T(G)$ must have exponent at most $2$ unless $G$ satisfies some fairly strong conditions. As applications of our main theorem, we are able to show that $T(G)$ has order $2$ for all almost simple groups $G$, and that $T(G)$ has exponent at most $2$ for all centerless perfect or complete groups $G$.

math.GR↗
arXiv2024

Normalizer Quotients of Symmetric Groups and Inner Holomorphs

We show that every finite group $T$ is isomorphic to a normalizer quotient $N_{S_n}(H)/H$ for some $n$ and a subgroup $H\leq S_n$. We show that this holds for all large enough $n\ge n_0(T)$ and also with $S_n$ replaced by $A_n$. The two main ingredients in the proof are a recent construction due to Cornulier and Sambale of a finite group $G$ with $\mathrm{Out}(G)\cong T$ (for any given finite group $T$) and the determination of the normalizer in $\mathrm{Sym(G)}$ of the inner holomorph $\mathrm{InHol}(G)\leq\mathrm{Sym}(G)$ for any centerless indecomposable finite group $G$, which may be of independent interest.

math.GR↗
arXiv2024

Classification of the types for which every Hopf--Galois correspondence is bijective

Let $L/K$ be any finite Galois extension with Galois group $G$. It is known by Chase and Sweedler that the Hopf--Galois correspondence is injective for every Hopf--Galois structure on $L/K$, but it need not be bijective in general. Hopf--Galois structures are known to be related to skew braces, and recently, the first-named author and Trappeniers proposed a new version of this connection with the property that the intermediate fields of $L/K$ in the image of the Hopf--Galois correspondence are in bijection with the left ideals of the associated skew brace. As an application, they classified the groups $G$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure on any $G$-Galois extension. In this paper, using a similar approach, we shall classify the groups $N$ for which the Hopf--Galois correspondence is bijective for every Hopf--Galois structure of type $N$ on any Galois extension.

math.NT↗
arXiv2024

On Gr\"{u}n's lemma for perfect skew braces

By previous work of Ced\'{o}, Smoktunowicz, and Vendramin, one already knows that the analog of Gr\"{u}n's lemma fails to hold for perfect skew left braces when the socle is used as an analog of the center of a group. In this paper, we use the annihilator instead of the socle. We shall show that the analog of Gr\"{u}n's lemma holds for perfect two-sided skew braces but not in general.

math.GR↗
arXiv2024

Representation theory of skew braces

According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility condition. Following their definition, we shall explain how some of the results from representation theory of groups, such as Maschke's theorem and Clifford's theorem, extend naturally to that of skew braces. We shall also give some concrete examples to illustrate that skew brace representations are more difficult to classify than group representations.

math.RT↗
arXiv2023

Finite almost simple groups whose holomorph contains a solvable regular subgroup

In our previous paper, we gave a complete list of the finite non-abelian simple groups whose holomorph contains a solvable regular subgroup. In this paper, we refine our previous work by considering all finite almost simple groups. In particular, our result yields a complete characterization of the finite almost simple groups which occur as the type of a Hopf-Galois structure on a solvable extension, or equivalently, the additive group of a skew brace having a solvable multiplicative group.

math.GR↗
arXiv2023

A generalization of Ito's theorem to skew braces

The famous theorem of Itô in group theory states that if a group $G=HK$ is the product of two abelian subgroups $H$ and $K$, then $G$ is metabelian. We shall generalize this to the setting of a skew brace $(A,{\cdot\,},\circ)$. Our main result says that if $A = BC$ or $A = B\circ C$ is the product of two trivial sub-skew braces $B$ and $C$ which are both left and right ideals in the opposite skew brace of $A$, then $A$ is meta-trivial. One can recover Itô's Theorem by taking $A$ to be an almost trivial skew brace.

math.GR↗
arXiv2023

Finite $p$-groups of class two with a small multiple holomorph

We consider the quotient group $T(G)$ of the multiple holomorph by the holomorph of a finite $p$-group $G$ of class two for an odd prime $p$. By work of the first-named author, we know that $T(G)$ contains a cyclic subgroup of order $p^{r-1}(p-1)$, where $p^r$ is the exponent of the quotient of $G$ by its center. In this paper, we shall exhibit examples of $G$ (with $r = 1$) such that $T(G)$ has order exactly $p-1$, which is as small as possible.

math.GR↗
arXiv2022

Finite $p$-groups of class two with a large multiple holomorph

Let $G$ be any group. The quotient group $T(G)$ of the multiple holomorph by the holomorph of $G$ has been investigated for various families of groups $G$. In this paper, we shall take $G$ to be a finite $p$-group of class two for any odd prime $p$, in which case $T(G)$ may be studied using certain bilinear forms. For any $n\geq 4$, we exhibit examples of $G$ of order $p^{n+{n\choose 2}}$ such that $T(G)$ contains a subgroup isomorphic to \begin{equation*} \operatorname{GL}_n(\mathbb{F}_p) \times \operatorname{GL}_{\binom{n}{2}-n}(\mathbb{F}_p).\end{equation*} For finite $p$-groups $G$, the prime factors of the order of $T(G)$ which are known so far all came from $p(p-1)$. Our examples show that the order of $T(G)$ can have other prime factors as well. In fact, we can embed any finite group into $T(G)$ for a suitable choice of $G$.

math.GR↗
arXiv2022

Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group

Let $G$ and $N$ be two finite groups of the same order. It is well-known that the existences of the following are equivalent: (a) a Hopf-Galois structure of type $N$ on any Galois $G$-extension; (b) a skew brace with additive group $N$ and multiplicative group $G$; (c) a regular subgroup isomorphic to $G$ in the holomorph of $N$. We shall say that $(G,N)$ is realizable when any of the above exists. Fixing $N$ to be a cyclic group, W. Rump (2019) has determined the groups $G$ for which $(G,N)$ is realizable. In this paper, fixing $G$ to be a cyclic group instead, we shall give a complete characterization of the groups $N$ for which $(G,N)$ is realizable.

math.GR↗