The classification of partition homogeneous groups with applications to semigroup theory

Abstract

Let λ = (λ 1 , λ 2 , . . .) be a partition of n, a sequence of positive integers in non-increasing order with sum n. Let Ω := {1, . . . , n}. An ordered partition P = (A 1 , A 2 , . . .) of Ω has type λ if |A i | = λ i .Following Martin and Sagan, we say that G is λ-transitive if, for any two ordered partitions P = (A 1 , A 2 , . . .) and Q = (B 1 , B 2 , . . .) of Ω of type λ, there exists g ∈ G with A i g = B i for all i. A group G is said to be λhomogeneous if, given two ordered partitions P and Q as above, inducing the setsThe first goal of this paper is to classify the λ-homogeneous groups (Theorems 1.1 and 1.2). The second goal is to apply this classification to a problem in semigroup theory.Let Tn and Sn denote the transformation monoid and the symmetric group on Ω, respectively. Fix a group H Sn. Given a non-invertible transformation a ∈ Tn \ Sn and a group G Sn, we say that (a, G) is an H-pair if the semigroups generated by {a} ∪ H and {a} ∪ G contain the same non-units, that is, a, G \ G = a, H \ H. Using the classification of the λ-homogeneous groups we classify all the Sn-pairs (Theorem 1.7). For a multitude of transformation semigroups this theorem immediately implies a description of their automorphisms, congruences, generators and other relevant properties (Theorem 8.5).This topic involves both group theory and semigroup theory; we have attempted to include enough exposition to make the paper self-contained for researchers in both areas.The paper finishes with a number of open problems on permutation and linear groups.

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

“…McAlister [80] proved that for any group G ≤ S n and any transformation a ∈ T n , the semigroups a, G \ G and g −1 ag | g ∈ G have the same idempotents. This proves (1).…”

“…(b) (Some) subgroups of PΓL(2, q) containing PSL(2, q), for q a prime power ("some" means "all" if and only if q is even or congruent to 3 mod 4). In fact, the numbers of orbits on the three types of partitions for G and its normalizer are (2, 10, 11) and (2,4,5) for G = AGL (1,8), and (2,5,11) and (2,4,10) for G = M 22 . Note that M 22 comes very close: only two orbits of each of the other two types are fused by M 22 : 2.…”

“…• G = N Sn (G), that is, • G = AGL (1,8), PGL (2,8), PGL (2,9), M 10 , PSL(2, 11), M 22 , PXL (2,25), or PXL (2,49), with k = 3 and λ = (4, 1, . .…”

A Complete Classification of Which (n, k)-Star Graphs are Cayley Graphs

“…(iii) H = AGL (1,8), AΓL (1,8) or AΓL (1,32). Then n = 8 or 32, so n − 1 = 7 or 31, which is prime in either case.…”

Primitive groups, graph endomorphisms and synchronization

“…Given the enormous progress made in the last three or four decades, permutation groups now has the tools to answer questions coming from the real world through transformation semigroups; these questions translate into beautiful statements in the language of permutation groups and combinatorial structures, as shown in many recent investigations (as a small sample, please see [2,4,6,7,8,12,13,20,29,39,42]).…”