{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,4]],"date-time":"2025-10-04T18:28:25Z","timestamp":1759602505784},"reference-count":17,"publisher":"Wiley","issue":"2","license":[{"start":{"date-parts":[[2006,10,3]],"date-time":"2006-10-03T00:00:00Z","timestamp":1159833600000},"content-version":"vor","delay-in-days":9620,"URL":"http:\/\/onlinelibrary.wiley.com\/termsAndConditions#vor"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Journal of Graph Theory"],"published-print":{"date-parts":[[1980,6]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Kronecker studied a class of groups \u3008<jats:italic>p, p<\/jats:italic> \u2010 1, <jats:italic>r<\/jats:italic>\u3009, whose commutator subgroups are prime cyclic of order <jats:italic>p<\/jats:italic>, and whose commutator quotient groups are cyclic of order <jats:italic>p<\/jats:italic> \u2010 1. These are now commonly called the <jats:italic>K<\/jats:italic>\u2010metacyclic groups. It follows from the classical work of Maschke that none of the <jats:italic>K<\/jats:italic>\u2010metacyclic groups except \u30083, 2, 2\u3009 has a planar Cayley graph. It is proved here that only for <jats:italic>p<\/jats:italic> = 5 and <jats:italic>p<\/jats:italic> = 7 is a <jats:italic>K<\/jats:italic>\u2010metacyclic group \u3008<jats:italic>p, p<\/jats:italic> \u2010 1, <jats:italic>r<\/jats:italic>\u3009 toroidal. To achieve this result, this paper develops a methodology for using Proulx's classification of toroidal groups by presentation to determine whether an explicitly given group is toroidal.<\/jats:p>","DOI":"10.1002\/jgt.3190040205","type":"journal-article","created":{"date-parts":[[2007,5,29]],"date-time":"2007-05-29T06:28:59Z","timestamp":1180420139000},"page":"165-172","source":"Crossref","is-referenced-by-count":12,"title":["A determination of the toroidal <i>k<\/i>\u2010metacyclic groups"],"prefix":"10.1002","volume":"4","author":[{"given":"Jonathan L.","family":"Gross","sequence":"first","affiliation":[]},{"suffix":"Jr.","given":"Samuel J.","family":"Lomonaco","sequence":"additional","affiliation":[]}],"member":"311","published-online":{"date-parts":[[2006,10,3]]},"reference":[{"key":"e_1_2_1_2_2","volume-title":"Ergebnisse der Mathematik und ihrer Grenzgebiete","author":"Coxeter H. S. 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