{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:44:10Z","timestamp":1753893850560,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Given a binary quasigroup $G$ of order $n$, a $d$-iterated quasigroup $G[d]$ is the $(d+1)$-ary quasigroup equal to the $d$-times composition of $G$ with itself. The Cayley table of every $d$-ary quasigroup is a $d$-dimensional latin hypercube. Transversals and diagonals in multiary quasigroups are defined so as to coincide with those in the corresponding latin hypercube.\r\nWe prove that if a group $G$ of order $n$ satisfies the Hall\u2013Paige condition, then the number of transversals in $G[d]$ is equal to $ \\frac{n!}{ |G'| n^{n-1}} \\cdot n!^{d}\u00a0 (1 + o(1))$ for large $d$, where $G'$ is the commutator subgroup of $G$. For a general quasigroup $G$, we obtain similar estimations on the numbers of transversals and near transversals in $G[d]$\u00a0 and develop a method for counting diagonals of other types in iterated quasigroups.<\/jats:p>","DOI":"10.37236\/9699","type":"journal-article","created":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T01:33:52Z","timestamp":1631842432000},"source":"Crossref","is-referenced-by-count":0,"title":["Transversals, Near Transversals, and Diagonals in Iterated Groups and Quasigroups"],"prefix":"10.37236","volume":"28","author":[{"given":"Anna A.","family":"Taranenko","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"23455","published-online":{"date-parts":[[2021,9,10]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i3p48\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v28i3p48\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,9,17]],"date-time":"2021-09-17T01:33:53Z","timestamp":1631842433000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v28i3p48"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,10]]},"references-count":0,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2021,7,1]]}},"URL":"https:\/\/doi.org\/10.37236\/9699","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2021,9,10]]},"article-number":"P3.48"}}