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\n Let\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n be a finite set such that\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n X<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n |X|=n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and let\n \n \n \n \n i<\/mml:mi>\n \n \u2a7d\n \n <\/mml:mo>\n j<\/mml:mi>\n \n \u2a7d\n \n <\/mml:mo>\n n<\/mml:mi>\n <\/mml:mrow>\n i\\leqslant j \\leqslant n<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . A group\n \n \n \n \n G<\/mml:mi>\n \n \u2a7d\n \n <\/mml:mo>\n \n \n S<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n G\\leqslant \\mathcal {S}_{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is said to be\n \n \n \n \n (<\/mml:mo>\n i<\/mml:mi>\n ,<\/mml:mo>\n j<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (i,j)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -homogeneous if for every\n \n \n \n \n I<\/mml:mi>\n ,<\/mml:mo>\n J<\/mml:mi>\n \n \u2286\n \n <\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n I,J\\subseteq X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , such that\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n I<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n i<\/mml:mi>\n <\/mml:mrow>\n |I|=i<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n J<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n j<\/mml:mi>\n <\/mml:mrow>\n |J|=j<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , there exists\n \n \n \n \n g<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n G<\/mml:mi>\n <\/mml:mrow>\n g\\in G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n I<\/mml:mi>\n g<\/mml:mi>\n \n \u2286\n \n <\/mml:mo>\n J<\/mml:mi>\n <\/mml:mrow>\n Ig\\subseteq J<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . (Clearly\n \n \n \n \n (<\/mml:mo>\n i<\/mml:mi>\n ,<\/mml:mo>\n i<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (i,i)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -homogeneity is\n \n \n \n i<\/mml:mi>\n i<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -homogeneity in the usual sense.)\n <\/p>\n

\n A group\n \n \n \n \n G<\/mml:mi>\n \n \u2a7d\n \n <\/mml:mo>\n \n \n S<\/mml:mi>\n <\/mml:mrow>\n \n n<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n <\/mml:mrow>\n G\\leqslant \\mathcal {S}_{n}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is said to have the\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -universal transversal property if given any set\n \n \n \n \n I<\/mml:mi>\n \n \u2286\n \n <\/mml:mo>\n X<\/mml:mi>\n <\/mml:mrow>\n I\\subseteq X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (with\n \n \n \n \n \n |<\/mml:mo>\n <\/mml:mrow>\n I<\/mml:mi>\n \n |<\/mml:mo>\n <\/mml:mrow>\n =<\/mml:mo>\n k<\/mml:mi>\n <\/mml:mrow>\n |I|=k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) and any partition\n \n \n \n P<\/mml:mi>\n P<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n of\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n into\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n blocks, there exists\n \n \n \n \n g<\/mml:mi>\n \n \u2208\n \n <\/mml:mo>\n G<\/mml:mi>\n <\/mml:mrow>\n g\\in G<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n such that\n \n \n \n \n I<\/mml:mi>\n g<\/mml:mi>\n <\/mml:mrow>\n Ig<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is a section for\n \n \n \n P<\/mml:mi>\n P<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . (That is, the orbit of each\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -subset of\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n contains a section for each\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -partition of\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .)\n <\/p>\n

\n In this paper we classify the groups with the\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -universal transversal property (with the exception of two classes of\n \n \n \n 2<\/mml:mn>\n 2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -homogeneous groups) and the\n \n \n \n \n (<\/mml:mo>\n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (k-1,k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -homogeneous groups (for\n \n \n \n \n 2<\/mml:mn>\n ><\/mml:mo>\n k<\/mml:mi>\n \n \u2a7d\n \n <\/mml:mo>\n \n \u230a\n \n <\/mml:mo>\n \n \n n<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n \n \u230b\n \n <\/mml:mo>\n <\/mml:mrow>\n 2>k\\leqslant \\lfloor \\frac {n+1}{2}\\rfloor<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ). As a corollary of the classification we prove that a\n \n \n \n \n (<\/mml:mo>\n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n ,<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n (k-1,k)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -homogeneous group is also\n \n \n \n \n (<\/mml:mo>\n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n ,<\/mml:mo>\n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n (k-2,k-1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -universal transversal property have the\n \n \n \n \n (<\/mml:mo>\n k<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n (k-1)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -universal transversal property.\n <\/p>\n

\n A corollary of all the previous results is a classification of the groups that together with any rank\n \n \n \n k<\/mml:mi>\n k<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n transformation on\n \n \n \n X<\/mml:mi>\n X<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n generate a regular semigroup (for\n \n \n \n \n 1<\/mml:mn>\n \n \u2a7d\n \n <\/mml:mo>\n k<\/mml:mi>\n \n \u2a7d\n \n <\/mml:mo>\n \n \u230a\n \n <\/mml:mo>\n \n \n n<\/mml:mi>\n +<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mfrac>\n \n \u230b\n \n <\/mml:mo>\n <\/mml:mrow>\n 1\\leqslant k\\leqslant \\lfloor \\frac {n+1}{2}\\rfloor<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ).\n <\/p>\n

The paper ends with a number of challenges for experts in number theory, group and\/or semigroup theory, linear algebra and matrix theory.<\/p>","DOI":"10.1090\/tran\/6368","type":"journal-article","created":{"date-parts":[[2015,7,1]],"date-time":"2015-07-01T08:51:23Z","timestamp":1435740683000},"page":"1159-1188","source":"Crossref","is-referenced-by-count":9,"special_numbering":"969","title":["Two generalizations of homogeneity in groups with applications to regular semigroups"],"prefix":"10.1090","volume":"368","author":[{"given":"Jo\u00e3o","family":"Ara\u00fajo","sequence":"first","affiliation":[]},{"given":"Peter","family":"Cameron","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2015,7,1]]},"reference":[{"issue":"7","key":"1","doi-asserted-by":"publisher","first-page":"1237","DOI":"10.1142\/S0218196711006923","article-title":"\ud835\udc63*-algebras, independence algebras and logic","volume":"21","author":"Ara\u00fajo, Jo\u00e3o","year":"2011","journal-title":"Internat. 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