Orbits of Primitive $k$-Homogenous Groups on $(n-k)$-Partitions with Applications to Semigroups

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 $k\ge 1$); 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.


Introduction
Let T n be the full transformation monoid on n points and let S n be its group of units, the symmetric group. The rank of a transformation t ∈ T n is the size of its image, and is denoted by rank(t). We say that a partition P = (A 1 , . . . , A k ), where the A i s are in a sequence of nonincreasing sizes, is of type (|A 1 |, . . . , |A k |). Given two sets A, B ⊆ T n , we denote by A, B the semigroup generated by A∪B; in case B = {t}, we will abuse notation writing A, t , rather than A, {t} .
In what group theory concerns, this paper investigates the minimal generating sets of some permutation groups G ≤ S n and their orbits on (n − k)-partitions (for k ≥ n/2). For example, in the particular case of k = 1, the general problem we are considering reads as the study of the orbits of G on (n − 1)-partitions, that is, the study of the orbits of G on 2-sets (a slight generalization of the key concept of orbitals). The main results for groups are of the following form (with m and m ′ appearing in several tables): Theorem 1.1. Let k ≤ n 2 and let G ≤ S n be a primitive k-homogenous group. Then, • G has m orbits on the set of (n − k)-partitions; • the smallest number of elements needed to generate G is m ′ .
A consequence of the previous result is that 2-homogeneous groups are 2-generated. That this result seems to have been unnoticed in the past, is perhaps the explanation for the fact that a very optimized algebra system such as GAP [45] provides sets of generators for the 2-transitive groups that in about 2/3 of the cases have more than 2 elements.
The next main group theory result, Theorem 3.2, builds upon the previous and provides a list of 3-homogeneous groups whose orbits on a given (n − 3)-partition coincide with those of their normalizers. (The list is complete except for one unresolved family.) In addition to dealing with natural questions on permutation groups, these results were crucial in order to generalize some semigroup theory results, as shown below in the sample Theorems 1.4 and 1.5, that we now introduce.
In his very influential and cited paper [80] McAlister proved the following.
Theorem 1.2. [80] Let G ≤ S n and t ∈ T n be any map of rank n − 1. Then G, t generates all rank n − 1 transformations in T n if and only if the group G has only one orbit on the (n−1)-partitions of {1, . . . , n}.
Another important result is due to Levi. Theorem 1.3. [60] Let A n ≤ G ≤ S n and let t ∈ T n \ S n . Then the automorphism group of G, t is isomorphic to S n .
The aim of this paper is to use the group theory results referred to above to generalize these results as follows.
Theorem 1.4. Let t be a singular map in T n , and suppose that t has kernel type (l 1 , . . . , l k ), with k ≥ n/2; let G be a group having only one orbit in the partitions of that type. Let S = t, G \ G. Then (2) if k ≤ n − 2, then G, t is generated by 3 elements; (3) let A be a set of rank k maps such that A, G generates all maps of rank at most k and A has minimum size among the sets with that property. Then |A| is given in Table 1.
In the previous theorem we require the group of units to be transistive on the kernel type of the singular map t. In the next theorem this condition is replaced by the weaker requirement of G to be |Xt|homogeneous.
Theorem 1.5. Let G be a primitive group with just one orbit on (n−k)sets, where 1 ≤ k ≤ n/2. Let t ∈ T n be a rank (n − k) map. Then (1) Aut ( G, t ) ∼ = N Sn ( G, t ).
(3) Let A ⊆ T n be a set of rank n − k maps such that G, A generates all maps of rank at most n−k, and suppose A has minimum size among the sets with that property. Then the size of A is bounded by the values in Table 2. p(k) S n , A n 2 Table 2.
Number of rank n − k maps needed to together with a k-homogeneous group G generate all the maps of rank not larger than n − k.
As said, the two results above are just sample theorems. For more detailed results we refer the reader to the sections below.
In what semigroup theory concerns, this paper belongs to the general area of investigating how the recent results on group theory, chiefly the classification of finite simple groups, can help the study of semigroups. (For other papers on this line of research, see for example [1,4,5,6,7,9,10,11,12,13,14,24,26,27,61,64,65,80,82,92] and the references therein.) The typical object in this field is a semigroup generated by a set of non-invertible transformations A ⊆ T n \ S n and a group of permutations G contained in S n . In this paper we are mainly concerned with the description of automorphisms and minimal generating sets, for semigroups having special given group of units.
If S is a semigroup and U is a subset of S, then we say that U generates S if every element of S is expressible as a product of the elements of U. The rank of a semigroup S, denoted by rank S, is the least number of elements in S needed to generate S. It is wellknown that a finite full transformation semigroup, on at least 3 points, has rank 3, while a finite full partial transformation semigroup, on at least 3 points, has rank 4 (see [50, Exercises 1.9.7 and 1.9.13]). The problem of determining the minimum number of generators of a semigroup is classical, and has been studied extensively; see, for example, [3,6,25,28,43,51,56,66,87] and the references therein. Given the importance of idempotent generated semigroups illustrated by the Erdos/Howie famous twin results (see [39,49] and also [2,23]) the related notion of idempotent rank appeared as natural and has also been widely investigated; the same can be said about the concepts of relative rank and nilpotent rank; see [22,29,37,41,46,47,48,63]. One of the goals of this paper is to contribute to this line of research.
Another classic topic in semigroup theory is the description of the automorphisms of semigroups. After the pioneer work of Schreier [89] and Mal'cev [76], proving that the group of automorphisms of T n is isomorphic to S n , a long sequence of new results followed (for example, [7,8,14,15,16,17,18,19,20,21,58,59,60,67,75,90,91,92,93] and the references therein). In addition to the general interest of studying automorphisms of mathematical structures, the description of automorphisms of semigroups turned out to be a key ingredient in Plotkin's universal algebraic geometry [84] and [31,32,33,40,57,77,78,79,85,86]. Here, we use the impressive progresses made in the theory of permutation groups during the last couple of decades, to contribute to this line of research by finding the automorphisms of semigroups with given group of units.
In Section 2 we prove the main theorems about the minimum number of generators of primitive groups, and we also give estimates on the number of orbits of primitive groups on (n − k)-partitions, for k ≥ n/2. In Section 3 we tackle the problem of independent interest of classifying the permutation groups in which all orbits on (n − k)-partitions are invariant under the normalizer. In Section 4 we apply the results proved in the previous sections to describe automorphisms and ranks of semigroups in which its group of units has just one orbit on the kernel type of t. In Section 5 we consider similar problems, but for semigroups whose group of units has just one orbit on the image of t. Section 6 contains some comments on the normalizers of 2-homogeneous or primitive groups. The paper ends with a section of open problems.

Group theory
The aim of this section is to prove all the results of this form.
Theorem 2.1. Let k ≤ n/2 and let G ≤ S n be a primitive k-homogenous group. Then, • G has m orbits on the set of (n − k)-partitions; • the smallest number of elements needed to generate G is m ′ .
The case in which least can be said is the case of k = 1. The rank r(G) of a transitive permutation group G (acting on {1, . . . , n}) is the number of G-orbits on ordered pairs from {1, . . . , n}. To handle the case of k = 1, we need a slightly different parameter, the number n 2 (G) of G-orbits on the set of 2-subsets of {1, . . . , n}. Clearly (r(G)−1)/2 ≤ n 2 (G) ≤ r(G) − 1; the lower bound holds when G has odd order (since then no pair of points can be interchanged by an element of G), and the upper bound when all the orbitals of G are self-paired. Note that r(G) ≤ n, with equality if and only if G is regular. In particular, a primitive group G has r(G) = n if and only if n is prime and G is cyclic of order n. We thus see that n 2 (G) ≤ n − 1 for transitive groups G; equality is realised for an elementary abelian 2-group acting regularly, but for primitive groups of degree greater than 2 we have n 2 (G) ≤ (n−1)/2, with equality only for the cyclic and dihedral groups of odd prime degree.
Theorem 2.2. Let G ≤ S n be a 1-homogeneous (that is, transitive) permutation group. Then • G has n 2 (G) orbits on the set of (n − 1)-partitions; • the smallest number of elements needed to generate G is at most Proof. The (n − 1)-partitions all have one part with two elements and all the other parts are singletons. Therefore the group has as many orbits on the (n − 1)-partitions as orbits on the set of 2-sets.
McIver and Neumann [81] showed that every subgroup of S n can be generated by ⌊n/2⌋ elements if n = 3, and by 2 if n = 3. This bound is best possible for arbitrary subgroups, but for transitive or primitive subgroups it has been improved in [71,72] to the statements in the theorem. Now we are going to prove that the minimum number of generators of any 2-homogeneous finite group is 2. (It is worth observing that we could not find this observation in the literature; we are grateful to Colva Roney-Dougal and Andrea Lucchini for independently confirming it.) The proof uses the following result proved by Lucchini and Menegazzo [70]. Here d(G) denotes the least number of elements of G needed to generate the whole G.  Proof. If G is almost simple, then it satisfies the conditions of the theorem, with M the simple socle. Since, in the case of socle PSL(d, q), the group G contains no graph automorphisms, we have d(G/M) ≤ 2 in all cases, so d(G) = 2.
If G is affine, its unique minimal normal subgroup M is elementary abelian, and the quotient H is a linear group; the relevant groups can be found in [34,38]. If the linear group has normal subgroup SL(d, q), Sp(d, q) (d > 1) or G 2 (q), then another application of Theorem 2.3 shows that d(H) = 2, whence d(G) = 2. For 1-dimensional semiaffine groups, the linear group is metacyclic, and the result is clear. The finitely many cases remaining can be dealt with case by case: in each case, explicit generators for the linear group are known, and where more than two are given it suffices to show that the corresponding linear group can be generated by two elements. The groups (apart from the sharply 2-transitive group of degree 59 2 with linear group SL(2, 5)×C 29 , which is clearly 2-generated), are within reach of GAP; the computation can be speeded up by taking the first potential generator to belong to a set of conjugacy class representatives.
Regarding the number of orbits on (n − k)-partitions, we start by the large values of k.
Theorem 2.5. Suppose that G is a permutation group of degree n which is k-homogeneous, where either 6 ≤ k ≤ n/2, or k = 5, n ≥ 25, or k = 4, n ≥ 34. Then: (1) G has p(k) orbits on the set of (n − k)-partitions, where p is the partition function; (2) there is one orbit on partitions of each possible type.
Proof. It follows from the Classification of Finite Simple Groups and known results about 4-and 5-homogeneous groups that any G under the assumptions of the theorem is S n or A n . The second assertion is a well-known fact about A n and S n . For the first, given a partition of {1, . . . , n} with n − k parts, for k ≤ n/2, subtracting one from the size of each part gives a partition of k, and every partition of k arises in this way; all set partitions of {1, . . . , n} corresponding to each fixed partition lie in the same orbit of the symmetric or alternating group.
The numbers of orbits of the finitely many k-homogeneous groups other than symmetric or alternating groups for k = 5 and k = 4 can be computed.
Theorem 2.6. Let G be a k-homogeneous group of degree n (where k = 4 or 5 and n ≥ 2k), other than S n or A n . Then the number of orbits of G on (n − k)-partitions are given in Tables 3 and 4

below.
Remark. We are grateful to Robin Chapman for independent confirmation of the values for PΓL (2,32) in Table 3.   Table 3. Orbits of 4-homogeneous groups on (n − 4)-partitions The situation is very different for the 2-and 3-homogeneous groups, to which we now turn. The main difference is that there are infinitely many such groups (apart from the symmetric and alternating groups), so there is no reason why the number of orbits on (n − k)-partitions should be bounded (and indeed it is not; it can grow as a polynomial in n, whose degree depends on k and on the partition considered).  Table 4. Orbits of 5-homogeneous groups on (n − 5)-partitions Proof. Since each 2-set lies in n − 2 sets of size 3, G has at most n − 2 orbits on 3-sets. Also, for any 2-set, there are at most n−2 2 2-sets disjoint from it, so there are at most this many orbits on (2, 2, 1, . . . , 1) partitions.
The bound on the number of orbits is best possible, as the next example shows.
Example 2.1. Let p be a prime congruent to −1 (mod 12). Let G be the group of order p(p − 1)/2 consisting of all maps of the field of integers mod p of the form x → ax + b, where a is a non-zero square. Its normalizer is the group of order p(p − 1), consisting of all maps of the above form for arbitrary non-zero p.
There is a theorem for 3-homogeneous groups similar to Theorem 2.7.
In fact we can say more. From CFSG, we know that, if G is 3homogeneous, then one of the following holds: • PSL(2, q) ≤ G ≤ PΓL(2, q), for some prime power q; • G = AGL(d, 2) for some d; • G is one of finitely many exceptions. In the first case, the order of G is O(n 3 ), so the number of orbits on partitions of shape 2 3 1 n−6 will be Ω(n 3 ). However, in the other two cases, the number of orbits is bounded by a constant, independent of d in the second case. This is clear for the third case, so consider the second. Suppose we have an (n − 3)-partition of {1, . . . , n}. Then the set of points lying in parts of size greater than 1 has cardinality at most 6, and so these points lie in an affine subspace of dimension at most 5. The group is transitive on affine subspaces of any given dimension, and the stabiliser of such a subspace has only a bounded number of orbits on its subsets of size at most 6. The number of orbits for this type can be calculated by looking at AGL (5,2). We find that the number of orbits on AGL(d, 2) on (n − 3)-partitions is 12 for d ≥ 5. The numbers of orbits on partitions of the different types is given in Table 5, with the same conventions as earlier.
3 Total 12 11 7 Table 5. Orbits of AGL(d, 2) on (n − 3)-partitions Similar data can be produced for any finite number of the other 3-homogeneous groups. Table 6 gives a selection of 3-homogeneous groups of degree n ≥ 7, which includes all the sporadic examples, all 4-homogeneous groups, and all examples with n ≤ 10.

Orbits of normalizers
In this section we will be interested in the following questions:  Table 6. Orbits of 3-homogeneous groups on (n − 3)-partitions (1) Given an orbit of the k-homogeneous group G on (n − k)partitions, what is the subgroup of the normalizer of G, in S n , which fixes that orbit? (The question is well-posed since N Sn (G)/G acts on the set of orbits.) (2) In particular, for which groups is it the case that every orbit on (n − k)-partitions is invariant under the normalizer, that is, the action of N Sn (G)/G on the set of orbits is trivial?
If G is the alternating group, then each of its orbits is stabilised by the symmetric group. For k ≥ 4, any other k-homogeneous group is equal to its normalizer, except for PGL (2,8) with n = 9. This group is 5-homogeneous, and so has the same orbits on partitions of type (5, 1, 1, 1, 1) as its normalizer. Computation shows that this is not the case for other types of 5-partitions. So we have the following theorem.
Theorem 3.1. Let k ≥ 4 and n ≥ 2k, and let G be a k-homogeneous group of degree n. Then G has the same orbits on (n − k)-partitions of any given type as its normalizer, except in the case of PGL (2,8), for which this assertion holds for partitions of type (5, 1, 1, 1, 1) but for no other types.
For q an odd prime power, q = p d , the quotient PΓL(2, q)/PSL(2, q) is isomorphic to C 2 × C d , where the factors are generated by a diagonal automorphism (an element of PGL(2, q) with non-square determinant) and the Frobenius field automorphism. If d is even (so that q is a square), this group contains three subgroups of index 2. Two of these are, respectively, PΣL(2, q), and the group generated by PGL(2, q) and the square of the Frobenius automorphism. 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 .) The main theorem of this section is the following.
Theorem 3.2. Suppose that G is a 3-homogeneous subgroup of S n , and λ a type of (n − 3)-partitions. Then (G, λ) is closed if one of the following holds: This theorem answers question (2) at the beginning of this section, with the exception of the groups PXL(2, q) referred to in its statement. Concerning question (1) the situation may be much more complicated as the following example shows.  Table 7 gives the number of G-orbits on the 14-partitions of various types, and the numbers with each of the three possible stabilisers.  For the group G = PSL(2, 2 p ), with p prime and p > 3, the situation is much simpler: no (2 p − 2) partition can be fixed by an element outside G, and so every orbit has stabiliser G. (This also shows, for example, that the numbers of orbits for PSL (2,32) are five times those for PΓL (2,32) given in Table 6.) We now give the proof of Theorem 3.2.
Proof . We begin by listing the 3-homogeneous groups.
These partitions correspond naturally to 4-subsets. Now orbits of PGL(2, q) on 4-tuples are parametrised by cross ratio: there is some flexibility about the definition, but I will assume that the cross ratio of (∞, 0, 1, a) is a. Now the 24 orderings of a 4-set give rise to a set of 6 cross ratios (or occasionally fewer) of the form So GF(q) \ {0, 1} is partitioned into sets of 6 (or fewer) corresponding to orbits of PGL(2, q) on 4-sets.
Now PΓL(2, q) is generated by PGL(2, q) and the Frobenius map x → x p , where q is a power of p, say q = p t . Thus there is a cyclic group of order t permuting the orbits (or the sets as above). To show that no proper subgroup of PΓL(2, q) containing PGL(2, q) is good, it suffices to find a 6-set which is fixed by no power of the Frobenius map except the identity.
Suppose that we have a 6-set {z, 1/z, 1 − z, 1/(1 − z), z/(z − 1), (z − 1)/z} which is fixed by a non-trivial power of the Frobenius map; we can assume that this has the form x → x p u where u divides t. We put t = uv and p u = r, so that q = r v and the map under consideration has fixed field GF(r). Now, for every z, z r ∈ {z, 1/z, 1 − z, 1/(1 − z), z/(z − 1), (z − 1)/z}. There are six possibilities: • z r = z. Then z ∈ GF(r).
Only one of these possibilities can hold. We may assume that q = r, so that z can be chosen so that the first possibility does not hold. Suppose that q = r 2 . Now the above argument shows that the r 2 − r elements outside GF(r) satisfy one of the three equations z r = 1/z, z r = 1 − z, or z r = z/(z − 1). These are polynomials of degrees r + 1, r, r + 1 respectively; so 3r + 2 ≥ r 2 − r, giving r ≤ 4. Now PGL(2, 4) ∼ = A 5 falls under case (a); and computation shows that (PGL (2,9), λ) is closed but PGL(2, 16) and PGL(2, 16) : 2 are not. (Both the last two groups have three orbits on 4-sets, but PGL(2, 16) : 4 has only two.) Now suppose that q = r 3 . We argue similarly to say that the elements outside GF(r) satisfy one of the two equations z r = 1/(1 − z) or z r = (z − 1)/z, both polynomials of degree r + 1. Thus r 3 − r ≤ 2(r + 1), with only the solution r = 2. The pair (PGL (2,8), λ) is good, since PGL(2, 8) is 4-homogeneous.
In this case, each orbit has a representative in which the 3-set is {∞, 0, 1}, by 3-transitivity of G. The elements of PGL(2, q) which map this set to itself are the maps z → f (z), where f (z) is one of the six linear fractional expressions which came up in our discussion of cross ratio. Moreover, all three points are fixed by the Frobenius map.
Suppose that p is odd. Take x ∈ GF(p) \ {0, 1} and y in no proper subfield of GF(q), and consider the partition as above whose 2-set is {x, y}.
The points fixed by the above transformations are x = −1, x = 2, x = 1 2 , and x a primitive 6th root of 1. If p = 3 or 7 we can choose x to satisfy none of these, so there is only one set in the orbit. But if we choose y to be a primitive element, then it is not fixed by any power of the Frobenius map, so this set is in a regular orbit of this map.
Suppose that p = 7. A similar but easier argument applies, since each of 2, 4 and 6 is fixed by just a single element, so we find r v − r ≤ r + 1, which is impossible.
Lastly we have the case p = 2. We may assume that t > 2, since PGL(2, 4) ∼ = A 5 . Now if y = 1/x, 1−x, x/(x−1), then only the identity in PGL(2, q) fixes this partition. Choosing y to be a primitive element of GF(q) shows that the group generated by the Frobenius map acts regularly on the orbit of this partition. Partitions of type (2, 2, 2, 1, . . . , 1).
We can assume that an orbit we are considering contains the partition {∞, 0}, {1, a} and {b, c}.
Suppose first that p > 2, and take a = 2. The three linear fractional transformations fixing the pair of sets making up the first two cycles are z → 2/z, z → (z −2)/(z −1), and z → 2(z −1)/(z −2), have among them at most six fixed points, namely ± √ 2, 1 ± √ −1, and 2 ± √ −2; so there is a point a fixed by none of these. If we take b and c to be linearly independent over GF(p), then only the identity in PGL(2, q) fixes the three sets; and if we take z = y p , then we find that they are not fixed by any power of the Frobenius map.
If p = 2, the argument is similar. If t is even, then we have a subfield GF(4); if t is divisible by 3, a subfield GF (8). If neither of these occurs, then no field automorphism can fix a partition of shape λ, since only permutations of order dividing 48 can do so. Groups not containing PGL(2, q).
If q is not a square, then any subgroup of PΓL(2, q) containing PSL(2, q) but not PGL(2, q) must lie inside PΣL(2, q). Moreover, we may assume that G = PΣL(2, q), since any other subgroup is contained in a group twice as large which does itself contain PGL(2, q). This case only arises if q ≡ 3 (mod 4), since otherwise PΣL(2, q) is not 3-homogeneous.
If the PΣL(2, q)-orbit of a partition P is fixed by PΓL(2, q), then a set in that orbit must be fixed by an element of PΓL(2, q) \ PΣL(2, q), since then its stabiliser will be twice as large, and the orbit the same size. We can assume that such an element has 2-power order, and all its cycles have the same size (since PΣL(2, q) has odd order). This excludes shape (3, 2, 1, . . . , 1), so we have to consider the other two types. Moreover, q is an odd power of p, so these maps do not involve field automorphisms.
First consider type (4, 1, . . . , 1), so we are looking for an element fixing a 4-set, acting on it as either a double transposition or a 4-cycle. So, if every orbit is accounted for, we have (q + 1)/4 ≤ 3, so q = 7 or q = 11. It can be checked (by hand or by computer) that PSL(2, 7) for type (4, 1, . . . , 1) is not closed, but PSL(2, 11) is. Now consider the type (2, 2, 2, 1, . . . , 1). This time we can assume that the partitioned 6-set is {{∞, 0}, {1, a}, {b, c}}, and it is fixed by an involution, which fixes one or all of the 2-sets. If there is a cycle (∞, 0), then 1 maps to a, b or c, and we find that the product of any two of a, b, c is the third. (For example, if (1, a) is a cycle, then the map is z → a/z, and a/b = c.) In the remaining case, we have four triple transpositions to consider, namely (∞, 1) (0, a) The remaining class to be considered are the groups PXL(2, q). As mentioned earlier, we have checked by computer the odd prime power squares up to 121, and found that PXL(2, q) acting on (4, 1, . . . , 1) partitions is closed for q = 9, 25 and 49, but not for q = 81 or 121.

Groups having only one orbit in a given kernel type
The remainder of this paper is dedicated to the application to semigroup theory of the results found above. We want to describe the structure (elements, ranks, automorphisms, congruences, regularity, idempotent generation, etc.) of semigroups generated by a k-homogenous subgroup of S n and some singular maps of rank larger than n/2. We will use several times the well known fact ( [50, p.11]) that if S is a finite semigroup and a ∈ S, then there exists a natural number ω such that a ω is idempotent.
In this section we are going to study the semigroups generated by a singular transformation t, such that rank(t) ≥ n/2, and a permutation group that has only one orbit on the kernel type of t.
We start by noting the following. Suppose that the kernel of t has type (l 1 , . . . , l k ) and m is the largest natural such that l m > 1. Then G must be ( m i=1 l i )-homogeneous and hence, given that the rank of t is k, the group must be p-homogeneous, for some p ∈ {n − k + 1, . . . , 2(n − k)}. The smallest value of p is attained if the kernel type is (n − k + 1, 1, . . . , 1), and the largest value is attained for kernel type (2, . . . , 2, 1, . . . , 1). Therefore, for any practical considerations we might assume that our groups are (n − k + 1)-homogeneous.
Theorem 4.1. Let t be a singular map in T (X), with X = {1, . . . , n}, and suppose that t has kernel type (l 1 , . . . , l k ), with k ≥ n/2; let G be a group having only one orbit in the partitions of that type. Let E denote the set of idempotents of t, G \ G. Then The proof of this theorem will follow from a sequence of lemmas. Throughout this section t ∈ T n will be a rank k map of kernel type (l 1 , . . . , l k ), and G ≤ S n will be a (n−k +1)-homogeneous group having only one orbit on the partitions of type (l 1 , . . . , l k ). We now introduce some notation. Given the rank and the kernel type of t we have where |A i | = l i (for all i ∈ {1, . . . , k}). Throughout this section we will assume that the fixed map t has kernel T = (A 1 , . . . A k ) of type (l 1 , . . . , l k ).
Observe that for every g, h ∈ G we have g −1 th = A 1 g · · · A k g a 1 h · · · a k h .
Since k ≥ n/2 and the group is (n−k +1)-homogeneous, it follows that the group is also k-homogeneous. Thus given any k-set Y contained in X, there exists th ∈ G, t such that Xth = Y . Similarly, given any partition Q = (B 1 , . . . , B k ) of X of type (l 1 , . . . , l k ), since G has only one orbit on the partitions of this type, it follows that there exists g ∈ G such that {A 1 , . . . , A k }g = {B 1 , . . . , B k } and hence the kernel of g −1 t is (B 1 , . . . , B k ). This proves the following lemma.
Lemma 4.2. Given any partition Q of type (l 1 , . . . , l k ) and any k-set Y ⊆ X, there exist g, h ∈ G such that ker(g −1 th) = Q and Xg −1 th = Y .
The previous result shows that G, t has rank k maps of every possible image and kernel. The next result provides the anlogous result for idempotents. Lemma 4.3. Given any partition Q of type (l 1 , . . . , l k ) and any k-set Y ⊆ X such that Y is a transversal for Q, there exists an idempotent e ∈ t, G such that ker(e) = Q and Xe = Y .
Proof . By the previous lemma we know that there exist g, h ∈ G such that ker(g −1 th) = Q and Xg −1 th = Y . Since Y is a transversal for Q, it follows that there exists k ∈ G, namely k := hg −1 , such that rank(tkt) = rank(t). Therefore, every element in tk := {(tk) i | i ∈ N}, the monogenic semigroup generated by tk, has the same rank as t. Since every finite semigroup contains an idempotent, we conclude that tk contains an idempotent, say (tk) ω , and hence g −1 (tk) ω g = g −1 (thg −1 ) ω g is also an idempotent with kernel Q and image Y . The result follows.
In order to increase the readability of the arguments we introduce some notation. Given a partition P = (A 1 , . . . , A k ) of X, and a transversal S = {a 1 , . . . , a k } for P , where a i ∈ A i (for all i), we represent A i by [a i ] P and the pair (P, S) induces an idempotent mapping defined by [a i ] P e = {a i }. Conversely, every idempotent can be so constructed from a partition and a transversal. With this notation, we can write the idempotent  Proof . By the previous lemma, given any partition of the same type as the kernel of t, and any transversal for it, there exists in t, G an idempotent with that partition as kernel and that transversal as image. Therefore we can pick a partition of the same type as the kernel of t with the following parts: Q 0 = {{b 1 , b k+1 , . . .}, {b 2 , . . .}, . . . , {b k , . . .}}; it is clear that Xq 2 is a transversal for Q 0 and hence the idempotent Since given any two k-sets Y, Z ⊆ X, there exists a sequence of ksubsets of X, say (Y 1 , . . . , Y m ), such that |Y i ∩Y i+1 | = k−1, with Y 1 = Y and Y m = Z, the following result is a consequence of the application of the previous lemma as many times as needed.
Corollary 4.5. Let q 1 , q 2 ∈ G, t be two rank k maps. Then there exists a sequence of idempotents e 1 , . . . , e j such that Xq 1 e 1 . . . e j = Xq 2 .
So far we showed that it is possible to use idempotents to build maps with any given kernel of the same type as the kernel of t, and as image any k-set. Now we move a step forward. Taking e 1 ∈ A, e 2 ∈ B, and e 3 ∈ C, all from G, t , we get the desired composition p(xy) = pe 1 e 2 e 3 . Now we can prove Theorem 4.1.
Proof . To prove the theorem, observe that t, S n \ S n is generated by the set {gth | g, h ∈ S n }. If gth ∈ t, G , for all g, h ∈ S n , the result would follow.
Let Q be the partition induced by the kernel of gth and let S 1 be a transversal for Q. Since Q has the same kernel type as ker(t) it follows, by Lemma 4.3, that there exists an idempotent e ∈ G, t such that Xe = S 1 and the kernel of e is Q. Let S be a transversal for the kernel of t. By Corollary 4.5, there exists a sequence of idempotents such that Xee 1 . . . e i = S. Thus the map ee 1 . . . e i t has the same rank as t, and the same kernel as gph. Similarly, there are idempotents f 1 , . . . , f l such that Xee 1 . . . e i tf 1 , . . . , f l = Xgth. Thus, there exists a permutation σ of the set Xgth such that ee 1 . . . e i tf 1 , . . . , f l σ = gth. Therefore, Xgth = ee 1 . . . e i tf 1 , . . . , f l σ = ee 1 . . . e i tf 1 , . . . , f l (x 1 y 1 ) . . . (x m y m ), and each of these transpositions can be replaced by a product of three idempotents of t, G . This also proves that t, G \ G ⊆ E, t ; as the converse inclusion is obvious, the result follows.
We recall here some known facts about the semigroups t, S n . (2) S is regular, that is, for all a ∈ S there exists b ∈ S such that a = aba; (3) S is generated by its idempotents; (4) S and g −1 ag | g ∈ S n have the same idempotents; (5) S = g −1 ag | g ∈ S n ; (6) the automorphisms of a, S n are those induced under conjugation by the elements of the normalizer of S in S n , Aut( a, S n ) ∼ = N Sn ( a, S n ); (7) we also have Aut( a, S n ) ∼ = S n ; (8) all the congruences of a, S n are described; (9) if e 2 = e ∈ a, S n , r := rank(e), then {f ∈ a, S n | ker(f ) = ker(e) and Ωf = Ωe} ∼ = S r .
(10) regarding principal ideals and Green's relations, for all a, b ∈ S, we have aS = bS ⇔ ker(a) = ker(b) Sa = Sb ⇔ Ωa = Ωb SaS = SbS ⇔ rank(a) = rank(b) (11) the minimum size of a generating set for a, G , for a ∈ T n \ S n , is 3. (12) the minimum size of a set A of rank k maps such that A, S n generates all maps of rank at most k is p(k).
Proof . Equality (1) was proved by Symons in [92]. Claims (2), (3) and (5) were proved by Levi and McFadden in [65]. Claim (4) was proved by McAlister in [80], and (together with (3)) it also implies (5). Claim (6) follows from the general result that every automorphism of a semigroup S ≤ T n containing all the constants is induced under conjugation by the normalizer of S in S n (see [90] and also [19,20]); since, by (1), the semigroups S n , a contain all the constants, the result follows. Claim (7) was proved by Symons in [92], but is also an easy consequence from (6). In [62] Levi described all the congruences of an S n -normal semigroup and hence described the congruences in S. Thus (8).
Statement (9) belongs to the folklore (see Theorem 5.1.4 of [44]). The results about principal ideals (10) were proved by Levi and McFadden in [65].
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 . Therefore, to generate all maps of rank k a necessary and sufficient condition is that there is in A one map of each kernel type, so that |A| = p(n). It is well known that the maps of rank k, for k > 1, generate all maps of smaller ranks. The result follows.
The previous results immediately imply the following.
Theorem 4.8. Let t be a singular map in T n , the full transformation monoid on Ω := {1, . . . , n}, and suppose that t has kernel type (l 1 , . . . , l k ), with k ≥ n/2; let G be a group having only one orbit in the partitions of that type. Let S = t, G \ G. Then (2) S is regular, that is, for all a ∈ S there exists b ∈ S such that a = aba; (3) S is generated by its idempotents; (4) S and g −1 ag | g ∈ G have the same idempotents; (10) the minimum size of a generating set for a, G is 3. (11) let A be a set of rank k maps such that A, G generates all maps of rank at most k and A has minimum size among the sets with that property. A bound for the size of the sets A ⊆ T n such that G, A generate all maps of rank k is given in Table 8. In the middle column is the type on which G has only one orbit.
Proof . Claims (1)-(5) and (7) any p(k) Table 8. Generating all maps of rank k Regarding claim (6), observe that by (1) the semigroup a, G contains all the constant maps and hence, by [90,Theorem 1], its automorphisms are where, for a given g ∈ S n , we have τ g : a, G → a, G defined by f τ g = g −1 f g. Note that a permutation g ∈ N Sn (G) normalizes a, G if and only if it normalizes G and a, G \ G. Thus the automorphisms of a, G are the maps induced under conjugation by the elements in the normalizer N Sn (G) that also normalize a, G \G. Since a, G \G = a, S n \S n it follows that every permutation of S n normalizes a, G \G. We conclude that the automorphisms of a, G are all the maps {τ g : a, G → a, G | g ∈ N Sn (G)}.
To prove that in fact we have Aut ( a, G \ G) ∼ = N Sn (G) we only need to observe that primitive groups have trivial center (that is, only the identity in G commutes with all other elements of G).

Groups with only one orbit on the image
We turn now to semigroups t, G , where G is transitive on the image of t (that is, G is (n − k)-homogeneous, where k ≥ n/2 is the rank of t).
Theorem 5.1. Let G be a primitive group with just one orbit on (n−k)sets, where 1 ≤ k ≤ n/2. Let t ∈ T n be a map of rank n − k. Then (1) G, t \ G and g −1 tg | g ∈ G have the same idempotents; (2) Aut ( G, t ) ∼ = N Sn ( G, t ).
(3) For k ≥ 3, the list of 3-homogeneous groups that satisfy is the following: The list is complete with the possible exception of the groups PXL(2, q) for q ≥ 169. (4) Let A ⊆ T n be a set of rank k maps such that A, G generates all maps of rank at most k and A has minimum size among the subsets of T n with that property. Then the maximum sizes that A can have is given in Table 9.
For more precise values depending on the group chosen, see the tables in Section 2 of the paper.
Proof. 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).
A transitive group G is said to synchronize a map t if the semigroup G, t contains a constant map (and hence, by transitivity, all constant maps). It is proved that primitive groups synchronize every singular map of rank at least n − 4 (see [5,10,88]). It is also known that p(k) S n , A n 2 Table 9.
Worst case scenario of the smallest number of rank n − k maps needed to together with a khomogeneous group G generate all the maps of rank at most n − k.
2-homogeneous groups, together with any singular map, generate all the constant maps ( [13,80]). Therefore, under the assumptions of the theorem, if the primitive group G has only one orbit on the k-sets, for n > k ≥ n/2, then G together with any rank k map t generates all the constants and hence the automorphisms of S := t, G are induced under conjugation by the elements in N Sn (S). This implies (2).
The more detailed description included in (3) follows from Theorem 3.2.
Regarding (4), we start by observing that all maps in G, t having the same rank of t, have also the same kernel type of t. Therefore, to generate all rank k maps with G and a set A of rank k maps, A must contain maps whose kernels form a transversal of the orbits of G on each kernel type. This necessary condition turns out to be sufficient for G, A to generate all transformations of rank at most k. In fact, given any k-partition P and any transversal S for P , there exists p ∈ A and g ∈ G such that P = ker(gp). In addition, since G has only one orbit on the k-sets, it follows that there exists h ∈ G such that the image of gph is S. Therefore, we infer that rank(phg) = rank(p); thus, every element in phg has the same rank of p and, for some natural number ω, we have that (phg) ω is idempotent and the same holds for e := g(phg) ω g −1 . In addition, ker(e) = P and the image of e coincides with the image of ph which is S. Since P and S were arbitrary, it follows that A, G contains all rank k idempotents of T n . It is well known ( [2]) that the rank k idempotents generate all maps of rank at most k and hence the result follows.

On normalizers of 2-homogeneous groups
By the main theorems of the two previous sections, to compute the automorphisms of G, t (with G and t under the assumptions of the theorems) it is necessary to know the normalizer of G, t in S n , which is contained in the normalizer of G. Therefore we provide here the normalizers of 2-homogeneous groups.
According to a theorem of Burnside, a 2-transitive group G has a unique minimal normal subgroup T , which is either elementary abelian or simple non-abelian. (If G is 2-homogeneous but not 2-transitive, it also has a unique minimal normal subgroup, which is elementary abelian.) Thus N Sn (G) ≤ N Sn (T ). So, to describe the normalizers of the 2-homogeneous groups G, we only need to look within the group N Sn (T )/T . Table 10 gives the structure of this quotient in the case when T is simple. In the table, G(r, s, p) denotes the group a, b | a r = b s = 1, b −1 ab = a p . In all rows of the table except the second and fourth, N Sn (G) = N Sn (T ). In the second and fourth rows, we have N Sn (G)/T ∼ = N N (T )/T (G/T ), and this quotient is computed in the metacyclic group G(r, s, p).
Now we consider the 2-homogeneous affine groups. In each case, the classification gives a subgroup H (not necessarily 2-homogeneous) which must be contained in G. The group H contains the translation group T of G, so G = T G 0 and H = T H 0 . Thus, as in the other case, we have N Sn (G) ≤ N Sn (H), so again we have to compute the normalizer within the group N Sn (H)/H ∼ = N S n−1 (H 0 )/H 0 . Table 11 gives the structure of this quotient group. The groups G(r, s, p) are the same as defined earlier. In all cases not shown in the table, the quotient is abelian, and so the normalizers of H and T coincide, and we have not listed them explicitly.
We have not attempted to make a similar classification of normalizers of primitive groups, since this problem is as difficult as finding normalizers of arbitrary transitive groups, as the following example shows.
Let m ≥ 3, and let K be a transitive group of degree k. Let G be the wreath product S m ≀ K in its power action of degree m k . Then G is primitive, and its normalizer in the symmetric group of degree m k is S m ≀ N S k (K).
The next problem looks within reachable boundaries. Problem 2. Prove for 2-homogeneous groups an analogous of Theorem 3.2.
Unlike the previous, the next problem is certainly extremely difficult.
Problem 3. Prove for primitive groups an analogous of Theorem 3.2.
The next problem was introduced in Section 3, but only some remarks were then done. A full solution is still out there.
Problem 4. Given an orbit of the k-homogeneous group G on (n − k)partitions, what is the subgroup of the normalizer of G, in S n , which fixes that orbit?
The results on normalizers of 2-homogeneous groups suggest the following generalization.
Problem 5. Let G be a family of primitive groups that has been classified (e.g., primitive groups of rank 3). Build for the groups in that family a table similar to Table 10, describing the normalizers of these groups in the symmetric groups of the same degree.
In order to give a sharper version of Theorem 5.1 (2), it would be useful to classify the primitive groups having a 2-homogeneous normalizer in S n . Problem 6. Classify the primitive groups G ≤ S n such that N Sn (G) is 2-homogeneous.
The next conjecture seems a little wild, but might be true.
Problem 7. Let G be a primitive group and t ∈ T n \ S n . Then all the automorphisms of G, t are induced (under conjugation) by the elements in N Sn ( G, t ).
If t has rank, at least, n−4, then we know (by [5]) that G, t contains all the constant maps and hence (by [90]) the conjecture above holds. Similarly, if t has rank at most 2, we know by [82] that again G, t contains all the constant maps and the conjecture holds as above. Also by [82] we know that there are primitive groups G and singular maps t such that G, t do not contain the constant maps. The question is: what happens with the automorphisms of those semigroups? We conjecture they are all induced by the elements of the normalizer (in S n ) of the semigroup, but have no proof.
As shown above, every 2-homogeneous group is 2-generated. However the GAP library of 2-transitive groups contains default sets of generators that in the major part of the cases have size larger than 2.
Problem 8. Produce a library of generating sets of size 2 for all the degree n and k-homogeneous primitive groups in the GAP library (for k ≥ 2).
Slightly connected to the previous problem is the following.
Problem 9. Include in GAP a very effective function to find the homogeneity of a given permutation group.
The next problem deals again with GAP libraries.
Problem 10. Let G be a k-homogeneous degree n primitive group in the GAP library of primitive groups. Produce a minimal set A of degree n transformations of rank k such that G, A generates all the transformation of rank at most k.
In [60] it is proposed the problem of finding the groups that can be the normalizers in S n of some semigroup S ⊆ T n . The main theorems of this paper provide some answers for that question, but we would like to propose the following conjecture.
Problem 11. Is it true that a group G is the normalizer in S n of a semigroup S ≤ T n if and only if G is the normalizer in S n of some group H ≤ S n ?
Our final problem asks for a sharper version of Theorem 5.1, (2).
Problem 12. For every pair (G, λ), where G ≤ S n is a k-homogeneous group and λ is an (n−k)-partition of n, classify the groups N Sn ( G, t ), where t is any map whose kernel has type λ.