Publication
Two generalizations of homogeneity in groups with applications to regular semigroups
| dc.contributor.author | Araújo, João | |
| dc.contributor.author | Cameron, Peter J. | |
| dc.date.accessioned | 2015-03-24T15:37:05Z | |
| dc.date.available | 2015-03-24T15:37:05Z | |
| dc.date.issued | 2014 | |
| dc.description.abstract | Let X be a finite set such that |X| = n and let i 6 j 6 n. A group G 6 Sn is said to be (i, j)-homogeneous if for every I, J ⊆ X, such that |I| = i and |J| = j, there exists g ∈ G such that Ig ⊆ J. (Clearly (i, i)-homogeneity is i-homogeneity in the usual sense.) A group G 6 Sn is said to have the k-universal transversal property if given any set I ⊆ X (with |I| = k) and any partition P of X into k blocks, there exists g ∈ G such that Ig is a section for P. (That is, the orbit of each k-subset of X contains a section for each k-partition of X.) In this paper we classify the groups with the k-universal transversal property (with the exception of two classes of 2-homogeneous groups) and the (k−1, k)-homogeneous groups (for 2 < k 6 ⌊n+12 ⌋). As a corollary of the classification we prove that a (k − 1, k homogeneous group is also (k − 2, k − 1)-homogeneous, with two exceptions; and similarly, but with no exceptions, groups having the k-universal transversal property have the (k − 1)-universal transversal property. A corollary of all the previous results is a classification of the groups that together with any rank k transformation on X generate a regular semigroup (for 1 6 k 6 ⌊n+1 2 ⌋). The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory. | por |
| dc.identifier.citation | Araújo João; Cameron, Peter J. - Two generalizations of homogeneity in groups with applications to regular semigroups. "Transactions of the American Mathematical Society" [Em linha]. ISSN 0002-9947 (Print) 1088-6850 (Online). (2014), p. 1-29 | por |
| dc.identifier.issn | 0002-9947 | |
| dc.identifier.issn | 1088-6850 | |
| dc.identifier.uri | http://hdl.handle.net/10400.2/3811 | |
| dc.language.iso | eng | por |
| dc.peerreviewed | yes | por |
| dc.subject | Transformation semigroups | por |
| dc.subject | Regular semigroups | por |
| dc.subject | Permutation groups | por |
| dc.subject | Primitive groups | por |
| dc.subject | Homogeneous groups | por |
| dc.title | Two generalizations of homogeneity in groups with applications to regular semigroups | por |
| dc.type | journal article | |
| dspace.entity.type | Publication | |
| oaire.citation.endPage | 29 | por |
| oaire.citation.startPage | 1 | por |
| oaire.citation.title | Transactions of the American Mathematical Society | por |
| person.familyName | Ribeiro Soares Gonçalves de Araújo | |
| person.givenName | João Jorge | |
| person.identifier.ciencia-id | EC1F-273A-9F24 | |
| person.identifier.orcid | 0000-0001-6655-2172 | |
| rcaap.rights | openAccess | por |
| rcaap.type | article | por |
| relation.isAuthorOfPublication | 1f7b349c-3251-480d-a3ac-e3cb4ef44f22 | |
| relation.isAuthorOfPublication.latestForDiscovery | 1f7b349c-3251-480d-a3ac-e3cb4ef44f22 |