Two generalizations of homogeneity in groups with applications to regular semigroups

Araújo, João; Cameron, Peter J‎.

doi:10.1090/tran/6368

Cited by 21 publications

(111 citation statements)

References 37 publications

(67 reference statements)

Supporting

Mentioning

110

Contrasting

Unclassified

Order By: Relevance

“…We use PXL(2, q) to denote the third subgroup of index 2, obtained by adjoining to PSL(2, q) the product of diagonal and Frobenius automorphisms. (When q = 9 this group is better known as M 10 , the point stabiliser in the Mathieu group M 11 . )…”

Section: Orbits Of Normalizersmentioning

confidence: 99%

See 1 more Smart Citation

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Araújo¹,

Bentz

Cameron

2018

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Let X be a finite set such that |X| = n, and let k < n/2. A group is k-homogeneous if it has only one orbit on the sets of size k. The aim of this paper is to prove some general results on permutation groups and then apply them to transformation semigroups. On groups we find the minimum number of permutations needed to generate k-homogeneous groups (for k ≥ 1); in particular we show that 2-homogeneous groups are 2-generated. We also describe the orbits of k-homogenous groups on partitions with n − k parts, classify the 3-homogeneous groups G whose orbits on (n − 3)-partitions are invariant under the normalizer of G in S n , and describe the normalizers of 2-homogeneous groups in the symmetric group. Then these results are applied to extract information about transformation semigroups with given group of units, namely to prove results on their automorphisms and on the minimum number of generators. The paper finishes with some problems on permutation groups, transformation semigroups and computational algebra.

show abstract

Section: Orbits Of Normalizersmentioning

confidence: 99%

“…Claim (11) follows from the fact that S n is generated by two elements. Regarding (12), observe that given any rank k map t we have that all rank k maps in t, G have the same kernel type as t; conversely, every rank k map of the same kernel type of p belongs to t, G .…”

Section: Groups Having Only One Orbit In a Given Kernel Typementioning

confidence: 99%

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Araújo¹,

Bentz

Cameron

2018

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Set u = gb ′′ , v = h. Then uv = gb ′′ h = a ′′ and vu = hgb ′′ = hgha ′′ g = ha ′′ g = b ′′ . Thus a ′′ ∼ p b ′′ , which proves (6).…”

Section: Conjugacy In Epigroups and Epigroup Elementsunclassified

“…The equivalence of (5) and (6) in Theorem 4.5 was proved for regular semigroups by Kudryavtseva [43, Cor. 6 and Thm.…”

Section: Conjugacy In Epigroups and Epigroup Elementsmentioning

confidence: 99%

Four notions of conjugacy for abstract semigroups

Araújo¹,

Kinyon²,

Konieczny³

et al. 2017

Proceedings of the Royal Society of Edinburgh: Section A Mathem

Self Cite

View full text Add to dashboard Cite

The action of any group on itself by conjugation and the corresponding conjugacy relation play an important role in group theory. There have been many attempts to find notions of conjugacy in semigroups that would be useful in special classes of semigroups occurring in various areas of mathematics, such as semigroups of matrices, operator and topological semigroups, free semigroups, transition monoids for automata, semigroups given by presentations with prescribed properties, monoids of graph endomorphisms, etc. In this paper we study four notions of conjugacy for semigroups, their interconnections, similarities and dissimilarities. They appeared originally in various different settings (automata, representation theory, presentations, and transformation semigroups). Here we study them in full generality. The paper ends with a large list of open problems.

show abstract

“…In many practical situations we know that a primitive group contains a given permutation and we want to know which group it can be; in some other practical situations we know the group and would like to know if it contains a permutation of some given type. For example, a group G ≤ S n is said to be nonsynchronizing if it is contained in the automorphism group of a non-trivial primitive graph with complete core, that is, with clique number equal to chromatic number (see for example, [1,2,3,5,6,9,30]).…”

Section: Introductionmentioning

confidence: 99%

Imprimitive permutations in primitive groups

et al. 2017

View full text Add to dashboard Cite

The goal of this paper is to study primitive groups that are contained in the union of maximal (in the symmetric group) imprimitive groups. The study of types of permutations inside primitive groups goes back to the origins of the theory of permutation groups. However, this is another instance of a situation common in mathematics in which a very natural problem turns out to be extremely difficult. Fortunately, the enormous progresses of the last few decades seem to allow a new momentum on the attack to this problem. In this paper we prove that there are infinite families of primitive groups contained in the union of imprimitive groups and propose a new hierarchy for primitive groups based on that fact. In addition to the previous results and hierarchy, we introduce some algorithms to handle permutations, provide the corresponding GAP implementation, solve some open problems, and propose a large list of open problems. Lisboa,

show abstract

Two generalizations of homogeneity in groups with applications to regular semigroups

Cited by 21 publications

References 37 publications

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Orbits of primitive $k$-homogenous groups on $(n-k)$-partitions with applications to semigroups

Four notions of conjugacy for abstract semigroups

Imprimitive permutations in primitive groups

Contact Info

Product

Resources

About