A remark on commutative di-associative loops
type: Item
instance of: scholarly article
title: A remark on commutative di-associative loops
zbMATH Open document ID: 0597.20065
DOI: 10.1007/BF01188065
author: Bernhard Ganter
author: Ulrike Pfüller
published in: Algebra Universalis
publication date: 1985-01-01T00:00:00Z
review text: We say that a loop is of ''exponent'' n if every two distinct non-zero elements generate an abelian subgroup of exponent n. If \(n=p_ 1^{\alpha_ 1}...p_ k^{\alpha_ k}\) with \(p_ 1,...,p_ k\) primes then there is a loop of order m and ''exponent'' n iff \(m=\prod^{k}_{i=1}(1+v_ i(p_ i-1))\) where for every i there exists a Steiner system with parameters 2, \(p_ i+1\), \(v_ i\). Now using a theorem of \textit{R. W. Wilson} [J. Comb. Theory, Ser. A 13, 246- 273 (1972; Zbl 0263.05015)]: Let p be a fixed prime. Say m is admissible if it is of the form \(m=v(p-1)+1\) where v-1\(\equiv 0(p)\) and \(v(v- 1)\equiv 0(p+1)\). Then, up to finitely many exceptions, a loop of order m and ''exponent'' p exists iff m is admissible.
Mathematics Subject Classification ID: 20N05
Mathematics Subject Classification ID: 05B07
zbMATH DE Number: 3961990
zbMATH Keywords: commutative di-associative loops
zbMATH Keywords: Steiner system
reviewed by: Václav Havel
MaRDI profile type: Publication
cites work: Q5606623
cites work: Vector loops
cites work: Near vector spaces over GF(q) and (v,q + 1,1) BIBDs
cites work: An existence theory for pairwise balanced designs. I: Composition theorems and morphisms
full work available at URL: https://doi.org/10.1007/bf01188065
OpenAlex ID: W2087432520
description: scientific article; zbMATH DE number 3961990