Related papers: Classification of braces of order $p^3$

In this article we classify the left braces of order $p^2q$ where $p,q$ are primes fulfilling $q > p+1$. This classification includes a proof of three conjectures of Guarnieri and Vendramin (\cite[Conjectures 6.2-6.4]{Vendramin_skew})…

We describe all left braces of size 8p for p an odd prime different from 3 or 7 and validate the number given by Bardakov, Neschadim and Yadav. We give a characterization for isomorphism classes of a semidirect product of left braces and…

This is the first part of a series of two articles. In this paper we enumerate and classify the left braces of size $p^2q$ where $p$ and $q$ are distinct prime numbers by the classification of regular subgroups of the holomorph of the…

We consider relatively prime integer numbers $m$ and $n$ such that each solvable group of order $mn$ has a normal subgroup of order $m$. We prove that each brace of size $mn$ is a semidirect product of a brace of size $m$ and a brace of…

In this paper, we study the irreducible representations of skew braces of order \( pq \), which is equivalent to studying the representation theory of groups of order \( p^2q^2 \) arising from skew left braces, where \( p > q \) are primes.…

Let $p$ be a prime number and let $n$ be an integer not divisible by $p$ and such that every group of order $np$ has a normal subgroup of order $p$. (This holds in particular for $p>n$.) We prove that left braces of size $np$ may be…

We improve Algorithm 5.1 of [Math. Comp. {\bf 86} (2017), 2519-2534] for computing all non-isomorphic skew left braces, and enumerate left braces and skew left braces of orders up to 868 with some exceptions. Using the enumerated data, we…

In a recent paper Y. Hu has given a sufficient condition for the fundamental group of the r-th cyclic branched covering of S^3 along a prime knot to be left-orderable in terms of representations of the knot group. Applying her criterion to…

The right conjugacy closed loops of order 2p, where p is an odd prime, are classified up to isomorphism.

This paper presents a classification of all p-groups of order p^5 up to isomorphism. It contains a full list of their polycyclic presentations, a short introduction to the basic ideas of the methodes used to classify the groups, and a…

We show that the number of isomorphism classes of left braces of order~$64$ with additive group isomorphic to $C_2\times C_2\times C_4\times C_4$ is $10\,326\,821$. This completes the classification of left braces of order~$64$, that turn…

Using only undergraduate-level methods, we classify all groups of order $p^4$, where $p$ is an odd prime.

We determine right nilpotency of braces of cardinality $p^4$. If a brace of cardinality $p^4$ has an abelian multiplicative group, then it is left and right nilpotent, so we only consider braces with non-abelian multiplicative groups. We…

Let $\mathbb{F}_{p^k}$ be a finite field of odd characteristic $p$. In this paper we give a classification, up to isomorphism, of the associative commutative $\mathbb{F}_{p^k}$-algebras, starting from the connection with their bi-brace…

We classify nilpotent pre-Lie rings of cardinality $p^4$ and thereby braces of the same cardinality, for a sufficiently large prime $p$. It has been shown that nilpotent pre-Lie rings of cardinality $p^n$ correspond to strongly nilpotent…

In this paper we study the $R$-braces $(M,+,\circ)$ such that $M\cdot M$ is cyclic, where $R$ is the ring of $p$-adic and $\cdot$ is the product of the radical $R$-algebra associated to $M$. In particular, we give a classification up to…

We describe all $\mathbb F_{p}$-braces of cardinality $p^{4}$ which are not right nilpotent. Our solution illustrates a general way of investigating $\mathbb F_{p}$-braces of cardinality $p^{n}$ with a given multiplicative group. The…

We classify all groups of order $p^5$ with non-trivial unramified Brauer groups. We show that if $p>3$, then there are precisely $\gcd (p-1,4)+\gcd (p-1,3)+1$ such groups.

We classify all skew braces of Heisenberg type for a prime number $ p>3 $. Furthermore, we determine the automorphism group of each one of these skew braces (as well as their socle and annihilator). Hence, by utilising a link between skew…

Let $n \geq 1$ be a squarefree integer, and let $M$, $A$ be two groups of order $n$. Using our previous results on the enumeration of Hopf-Galois structures on Galois extensions of fields of squarefree degree, we determine the number of…