Utilize este identificador para referenciar este registo: http://hdl.handle.net/10400.2/3811
Título: Two generalizations of homogeneity in groups with applications to regular semigroups
Autor: Araújo, João
Cameron, Peter J.
Palavras-chave: Transformation semigroups
Regular semigroups
Permutation groups
Primitive groups
Homogeneous groups
Data: 2014
Citação: 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
Resumo: 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.
Peer review: yes
URI: http://hdl.handle.net/10400.2/3811
ISSN: 0002-9947
1088-6850
Aparece nas colecções:Matemática e Estatística - Artigos em revistas internacionais / Papers in international journals

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