Automorphic loops and metabelian groups

The free commutative automorphic $2$-generated loop of nilpotency class $3$

A loop is automorphic if all its inner mappings are automorphisms. We construct the free commutative automorphic $2$-generated loop of nilpotency class $3$. It has dimension $8$ over the integers.

素幂阶自守环的幂零性

Nilpotency in automorphic loops of prime power order

一个循环是自同构的，如果它的内部映射是自同构。使用所谓的相关运算，我们证明了每一个奇素数幂阶的交换自守环都是中心幂零的。从素数阶$p$域上的$2\times 2$矩阵向量空间中的各向异性平面出发，构造了一族具有平凡中心的$p^3$阶自守环.

Automorphisms of dihedral-like automorphic loops

Automorphic loops are loops in which all inner mappings are automorphisms. A large class of automorphic loops is obtained as follows: Let $m$ be a positive even integer, $G$ an abelian group, and $α$ an automorphism of $G$ that satisfies $α^2=1$ if $m>2$. Then the dihedral-like automorphic loop $\mathrm{Dih}(m,G,α)$ is defined on $\mathbb Z_m\times G$ by $(i,u)(j,v)=(i+j, ((-1)^{j}u+v)α^{ij})$. We prove that two finite dihedral-like automorphic loops $\mathrm{Dih}(m,G,α)$, $\mathrm{Dih}(\overline{m},\overline{G},\overlineα)$ are isomorphic if and only if $m=\overline{m}$, $G=\overline{G}$, and…

Half-isomorphisms of automorphic loops

Automorphic loops are loops in which all inner mappings are automorphisms. This variety of loops includes groups and commutative Moufang loops. A half-isomorphism $f : G \longrightarrow K$ between multiplicative systems $G$ and $K$ is a bijection from $G$ onto $K$ such that $f(ab)\in\{f(a)f(b), f(b)f(a)\}$ for any $a,b\in G$. A half-isomorphism is trivial when it is either an isomorphism or an anti-isomorphism. Consider the class of automorphic loops such that the equation $x\cdot(x\cdot y) = (y\cdot x)\cdot x $ is equivalent to $x\cdot y = y\cdot x$. Here we show that this class of loops incl…

Commutative automorphic loops of order $p^3$

A loop is said to be automorphic if its inner mappings are automorphisms. For a prime $p$, denote by $\mathcal A_p$ the class of all $2$-generated commutative automorphic loops $Q$ possessing a central subloop $Z\cong \mathbb Z_p$ such that $Q/Z\cong\mathbb Z_p\times\mathbb Z_p$. Upon describing the free $2$-generated nilpotent class two commutative automorphic loop and the free $2$-generated nilpotent class two commutative automorphic $p$-loop $F_p$ in the variety of loops whose elements have order dividing $p^2$ and whose associators have order dividing $p$, we show that every loop of $\math…

Automorphic loops arising from module endomorphisms

A loop is automorphic if all its inner mappings are automorphisms. We construct a large family of automorphic loops as follows. Let $R$ be a commutative ring, $V$ an $R$-module, $E=\mathrm{End}_R(V)$ the ring of $R$-endomorphisms of $V$, and $W$ a subgroup of $(E,+)$ such that $ab=ba$ for every $a$, $b\in W$ and $1+a$ is invertible for every $a\in W$. Then $Q_{R,V}(W)$ defined on $W\times V$ by $(a,u)(b,v) = (a+b,u(1+b)+v(1-a))$ is an automorphic loop. A special case occurs when $R=k<K=V$ is a field extension and $W$ is a $k$-subspace of $K$ such that $k1\cap W = 0$, naturally embedded into $\…

交换自守环的构造

Constructions of commutative automorphic loops

内部映射是自同构的循环是\emph{automorphic loop}（或\emph{A-loop}）。我们刻画了具有指数为2的中间核的交换（A-）回路，并解决了同构问题。利用这一特征和某些中心的三线性形式的基础上，我们构造了几类交换A-循环的2次幂。我们开始分类的交换A-回路的小订单，也有订单$p^3$，其中$p$是一个素数。

Enumeration of involutory latin quandles, Bruck loops and commutative automorphic loops of odd prime power order

There is a one-to-one correspondence between involutory latin quandles and uniquely $2$-divisible Bruck loops. Bruck loops of odd prime power order are centrally nilpotent. Using linear-algebraic approach to central extensions, we enumerate Bruck loops (and hence involutory latin quandles) of order $3^k$ for $k\le 5$, except for those loops that are central extensions of the cyclic group of order $3$ by the elementary abelian group of order $3^4$. Among the constructed loops there is a Bruck loop of order $3^5$ whose associated $Γ$-loop is not a commutative automorphic loop. We independently e…

交换自守环的结构

The Structure of Commutative Automorphic Loops

\emph{automorphic loop}（或\emph{A-loop}）是一个循环，其内部映射是自同构。交换A-环的每个元素生成一个群，并且$(xy)^{-1} = x^{-1}y^{-1}$成立。设$Q$是有限交换A-圈，$p$是素数.循环$Q$的阶为$p$的幂，当且仅当$Q$的每个元素的阶为$p$的幂。循环$Q$分解为奇数阶循环和2的幂阶循环的直积。如果$Q$是奇数阶，则它是可解的。如果$A$是$Q$的一个子环，则$|A|$可分为$|Q|$。如果$p$整除$|Q|$，则$Q$包含顺序为$p$的元素。如果有一个有限的简单非结合交换A-环，它的指数为2。

一类范畴同构于奇数阶Bruck环的环

A class of loops categorically isomorphic to Bruck loops of odd order

我们定义了一种新的循环，我们称之为$Γ$ -循环。在证明了$Γ$ -循环是幂结合的之后，我们的主要目标将是证明奇数阶的Bruck循环和奇数阶的$Γ$ -循环之间的范畴同构。一旦建立了这一点，我们就可以使用奇数阶Bruck环的结构来导出奇数阶$Γ$ -环的奇数阶、Lagrange和Cauchy定理，以及有限$Γ$ - $p$ -环（$p$奇）的中心的非平凡性。最后，我们回答了一个问题所提出的Jedlička，Kinyon和Vojtenchovskiy关于存在霍尔$π$ -子回路和Sylow $p$ -子回路交换自守循环。通过证明交换自守环是$Γ$ -环，并利用范畴同构，我们给出了肯定的回答。

Supernilpotent groups and $3$-supernilpotent loops

We find a short equational basis for the variety of $3$-supernilpotent loops. We also present a conceptually simple proof that $k$-nilpotence and $k$-supernilpotence are equivalent for groups. Connections between $3$-supernilpotent loops, Moufang loops, code loops, automorphic loops and AIM loops are explored.

The multiplicative loops of Jha-Johnson semifields

The multiplicative loops of Jha-Johnson semifields are non-automorphic finite loops whose left and right nuclei are the multiplicative groups of a field extension of their centers. They yield examples of finite loops with non-trivial automorphism group and non-trivial inner mappings. Upper bounds are given for the number of non-isotopic multiplicative loops of order $q^{nm}-1$ that are defined using the twisted polynomial ring $K[t;σ]$ and a twisted irreducible polynomial of degree $m$, when the automorphism $σ$ has order $n$.

On connected quandles of prime power order

We develop some general ideas to study connected quandles of prime power size and we classify non-affine connected quandles of size $p^3$ for $p>3$, using a combination of group theoretical and universal algebraic tools. As a byproduct we obtain a classification of Bruck loops of the same size and that they are in one-to-one correspondence with commutative automorphic loops.

有限Bruck环

Finite Bruck Loops

一个循环$(X,\circ)$被称为Bruck循环，如果它满足（右）Bol恒等式$((z\circ x)\circ y)\circ x = z\circ ((x\circ y)\circ x)$和自守逆性质$(x\circ y)^{-1}=x^{-1}\circ y^{-1}$。若$X$是有限Bruck环，$G$是由所有右平移$R(x): y\mapsto y\circ x$生成的群，则证明了$X$和$G$是中心积$X = O^{2'}(X) * O(X)$和$G = O^{2'}(G) * O(G)$，其中$O^{2'}(X)$（$O^{2'}(G)$）是由所有2-元生成的子环（子群），$O(X)$（$O(G)$）是奇阶的最大正规子环（子群）.特别地，如果$X$是可解的，那么这些中心积是直积。我们还给出了一组必要条件，必须持有有限布鲁克循环$X$是不可解的，但有每个正常的部分可解的，特别是，$X$必须是简单的，由2-元素，而商$G$由其最大的正常2-子群必须同构于$PGL_2(q)$，与$q=2^n+1\geq 5$。