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Two generalizations of homogeneity in groups with applications to regular semigroups


Authors: João Araújo and Peter J. Cameron
Journal: Trans. Amer. Math. Soc. 368 (2016), 1159-1188
MSC (2010): Primary 20B30, 20B35, 20B15, 20B40, 20M20, 20M17
Published electronically: July 1, 2015
MathSciNet review: 3430360
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Abstract: Let $ X$ be a finite set such that $ \vert X\vert=n$ and let $ i\leq j \leq n$. A group $ G\leq \mathcal {S}_{n}$ is said to be $ (i,j)$-homogeneous if for every $ I,J\subseteq X$, such that $ \vert I\vert=i$ and $ \vert J\vert=j$, there exists $ g\in G$ such that $ Ig\subseteq J$. (Clearly $ (i,i)$-homogeneity is $ i$-homogeneity in the usual sense.)

A group $ G\leq \mathcal {S}_{n}$ is said to have the $ k$-universal transversal property if given any set $ I\subseteq X$ (with $ \vert I\vert=k$) and any partition $ P$ of $ X$ into $ k$ blocks, there exists $ g\in 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\leq \lfloor \frac {n+1}{2}\rfloor $). 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\leq k\leq \lfloor \frac {n+1}{2}\rfloor $).

The paper ends with a number of challenges for experts in number theory, group and/or semigroup theory, linear algebra and matrix theory.


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Additional Information

João Araújo
Affiliation: Universidade Aberta and Centro de Álgebra, Universidade de Lisboa, Av. Gama Pinto, 2, 1649-003 Lisboa, Portugal
Email: jaraujo@ptmat.fc.ul.pt, jjrsga@gmail.com

Peter J. Cameron
Affiliation: Department of Mathematics, School of Mathematical Sciences at Queen Mary, University of London, London E1 4NS, United Kingdom
Email: pjc20@st-andrews.ac.uk

DOI: https://doi.org/10.1090/tran/6368
Keywords: Transformation semigroups, regular semigroups, permutation groups, primitive groups, homogeneous groups
Received by editor(s): April 10, 2012
Received by editor(s) in revised form: December 11, 2012, and December 18, 2013
Published electronically: July 1, 2015
Article copyright: © Copyright 2015 American Mathematical Society