A remark on commutative di-associative loops
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.
scientific article; zbMATH DE number 3961990