In this section we are going to expand on the detection of the primitive, synchro- nizing, spreading or separating properties using functions. The first, motivating result is Rystsov’s Theorem (Theorem 2.8), which we reformulate here:
Theorem 8.1. Let Gbe a transitive permutation group onΩ, with |Ω|=n. The
following are equivalent: (a) Gis primitive;
(b) for any functionf: Ω→Ωwhose image has cardinalityn−1, the semigroup generated byGandf contains a constant function;
(c) for any idempotent function f: Ω →Ωwhose image has cardinality n−1, the semigroup generated byGandf contains a constant function.
Rystsov does not, in fact, explicitly state the above theorem. But in [91] he proved that if a 6= b are elements of Ω, then the orbital digraph corresponding to (a, b) is connected if and only ifG synchronizes the rank n−1 idempotent e defined by
xe= (
x ifx6=a; b if x=a.
Theorem 8.1 is an immediate consequence of this result, Higman’s characteriza- tion of primitivity in terms of the connectivity of orbital digraphs and the easy observation that ifGis a transitive group andf is any rankn−1 mapping, then hG∪ficontains a rankn−1 idempotent.
Theorem 8.2. Let Gbe a permutation group on Ω, with |Ω|=n. The following
are equivalent:
(a) Gis2-homogeneous;
(b) for any functionf: Ω→Ωwhose image has sizen−1, the semigroup gen- erated byGandf contains all transformations which are not permutations; (c) for any idempotent function f: Ω → Ω whose image has size n−1, the semigroup generated byGandf contains all transformations which are not permutations.
Proof. It is obvious that (b) implies (c).
To prove that (a) implies (b), let Gbe 2-homogeneous and let f: Ω →Ω be a rankn−1 map, whose unique non-singleton kernel class is {a, b}, and {a0} = Ω\Ωf. We claim thathG, ficontains all idempotents of rankn−1. In fact, ifeis one such idempotent, with non-singleton kernel class{c, d}and{c0}= Ω\Ωe, there existh, g∈Gsuch that{c, d}g={a, b}anda0h=c0. Therefore, the unique non- singleton kernel class ofgf his{c, d}and Ωgf h= Ωe; so gf hhas the same image and kernel ase. Sinceeis idempotent, it follows that its image is a section of its kernel, and hence the same happens withgf h; thus rank(gf h)k= rank(e) =n−1, for all natural numbersk; now we can partition the natural number as follows: for
all naturaliandj, we say thati∼jif (gf h)i= (gf h)j. By Schur’s Theorem [94],
there exists a part in this partition containinga,banda+b, that is, (gf h)a = (gf h)a+b= (gf h)a(gf h)b= (gf h)a(gf h)a= ((gf h)a)2,
and hence there exists a natural numberasuch that (gf h)a is idempotent, having
the same kernel and image as e. But this forces (gf h)a =e (because there is a
unique idempotent with given image and kernel), and the claim is proved. It is well known, see [64], that the rankn−1 idempotent maps generate all non-invertible maps and this concludes the proof of the implication.
To prove that (c) implies (a) suppose e is an idempotent with non-singleton kernel class{a, b}such thathf, Gicontains all non-invertible maps. Letf0: Ω→Ω
be a rankn−1 map with non-singleton kernel class{c, d}. Since, by hypothesis, f0∈ he, Gi, it follows thatf0 =g1eg2. . . egk, and hence (c, d)∈ker(g1eg2. . . egk).
As rank(g1eg2. . . egk) = rank(e) it follows that ker(g1eg2. . . egk) = ker(g1e); thus there existsg1 ∈Gsuch that{c, d}g1 ={a, b}. As {c, d} was an arbitrary 2-set, it follows thatGhas only one orbit on 2-sets. The implication follows.
The above theorem was first proved by McAlister [81].
Denote by Unif(Ω) the set of functionsf: Ω →Ω whose kernel is a uniform partition with at least two parts. The next result provides some characterizations of synchronizing groups.
Theorem 8.3. Let Gbe a transitive permutation group onΩ, with |Ω|=n. The
following are equivalent:
(a) there is no non-trivial partition P and setA such that Ag is a section for P, for allg∈G;
(b) for any functionf: Ω→Ωwhich is not a permutation, the semigroup gen- erated byGandf contains a constant function;
(c) for any idempotent function f: Ω → Ω which is not a permutation, the semigroup generated byGandf contains a constant function;
(d) for all t ∈ Unif(Ω), there exists a part A of Ker(t) and g ∈ G such that |Ωtg∩A|>1.
Proof. The equivalence (a) and (b) is the content of Theorem 3.8; the equivalence of (b) and (c) is immediate since every mapping has an idempotent positive power. The implication (d) implies (b) is essentially the content of Corollary 3.10: if G is not synchronizing, then a minimal rank mapping t not synchronized byG belongs to Unif(Ω). By (d), there exist g∈ Gand a part A of Ker(t) such that |Ωtg∩A|>1. Therefore rank(tgt)<rank(t), a contradiction to the choice oft.
Conversely, suppose that t ∈ Unif(Ω) is such thathG, ti contains a constant mapping. Then there existsg∈Gsuch that rank(tgt)<rank(t), which is equiva- lent to saying that, for some partAof Ker(t), we have|Ωtg∩A|>1. The result follows.
The following result provides a characterization of separation that parallels the equivalence of (a) and (d) in the previous result.
Theorem 8.4. Let Gbe a transitive permutation group onΩ, with |Ω|=n. The
following are equivalent: (a) Gis separating;
(b) for allt ∈Unif(Ω) and all parts A of Ker(t), there exists g ∈ Gsuch that |A∩Ωtg|>1.
Proof. Suppose thatGis separating, and lett∈Unif(Ω) be singular. PutB= Ωt and letAbe an arbitrary Ker(t)-class. Then|A| · |B|=|Ω|(becauset is uniform) and there existsg∈Gsuch that|A∩Bg|>1, asGis separating and the average value of|A∩Bg|is 1 by Theorem 5.12.
Conversely, suppose that Gis not separating. Then we have two non-trivial subsetsA, B of Ω such that |A| · |B| =|Ω| and |A∩Bg| = 1 for allg ∈ G. Let t∈Unif(Ω) be any mapping such thatA is a part of Ker(t) and Ωt=B. Then, by (b), there existsg∈Gsuch that|A∩Bg|>1, a contradiction.
The above theorem, in light of Theorem 8.3, provides another proof that sep- arating groups are synchronizing.
Theorem 5.13 gave a characterization of spreading groups. We close this section with another.
Theorem 8.5. Let G≤Sn be a transitive group acting on Ω. The following are
equivalent:
(a) for all proper subsets A of Ωand singular mappings t∈T(Ω), there exists g∈Gsuch that|Agt−1|>|A|;
(b) for all proper subsets A of Ω and idempotent singular mappings e∈T(Ω), there existsg∈Gsuch that|Age−1|>|A|.
Proof. That (a) implies (b) is obvious. Conversely, lett be a singular mapping on Ω. Then there exists a singular idempotent mappingeand h∈Sn such that
h−1e = t. By (b), there exists g ∈ G such that |Age−1| > |A|. Therefore, |Agt−1|=|Age−1h|=|Age−1|>|A|, as required.
Observe that synchronizing groups can be defined in terms of functions or in terms of section-regular partitions. The previous result, allowing a definition of spreading groups in terms of idempotents, leads to a parallel definition in terms of partitions and sections. A groupGis spreading if and only if, for everyA(Ω, and every partitionP of Ω with sectionB, there existsg∈Gsuch that|Ag∗B0|>|A|, whereB0 is the multiset with supportB that givesx∈B multiplicity the size of its part inP.