Automorphic loops arising from module endomorphisms

作者：Grishkov Alexander^*; Rasskazova Marina; Vojtechovsky Petr

来源：Publicationes Mathematicae-Debrecen, 2016, 88(3-4): 287-303.

DOI：10.5486/PMD.2016.7283

• automorphic loop
• automorphic loop of order p(3)
• automorphism group
• semidirect product
• field extension
• quadratic extension
• module
• module endomorphism

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 = End(R)(V) the ring of R-endomorphisms of V, and W a subgroup of (E,+) such that ab = ba for every a, b is an element of W and 1 + a is invertible for every a is an element of W. Then Q(R,V)(W) defined on W x 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 boolean AND W = 0, naturally embedded into End(k)(K) by a bar right arrow M-a, bM(a) = ba. In this case we denote the automorphic loop Q(R,V)(W) by Q(k<K)(W).
We call the parameters tame if k is a prime field, W generates K as a field over k, and K is perfect when char(k) = 2. We describe the automorphism groups of tame automorphic loops Q(k<K) (W), and we solve the isomorphism problem for tame automorphic loops Q(k<K)(W). A special case solves a problem about automorphic loops of order p(3) posed by Jedlicka, Kinyon and Vojtechovsky.
We conclude the paper with a construction of an infinite 2-generated abelian-by-cyclic automorphic loop of prime exponent.

• 出版日期2016