Chapter II: P -schemes
2.4 Orbit P -schemes and m -schemes
An important family ofm-schemes called orbit schemes, or what we callorbitm- schemes, was proposed and studied in [IKS09]. The blocks of suchm-schemes are orbits of group actions.
Definition 2.17 (orbit m-scheme [IKS09]). Given a finite set S, m ∈ N+, and a groupK ⊆ Sym(S)acting naturally onK, for eachk ∈ [m], define the partition Pk ofS(k) to be the partition into K-orbits with respect to the diagonal action of
K onS(k). Them-collectionΠ ={P1, . . . , Pm}is called theorbitm-schemeonS associated with the groupK.
This is indeed anm-scheme:
Theorem 2.2([IKS09]). Them-collectionΠin Definition 2.17 is anm-scheme on S.
We define orbitP-schemes in a similar way, except that the subgroupKofSym(S) is now replaced with a subgroup ofG, and the diagonal actions onS(k),k ∈[m]are replaced with the actions on right coset spaces by inverse right translation.
Definition 2.18(orbitP-scheme). LetP be a subgroup system over a finite group G, and letKbe a subgroup ofG. ForH ∈ P, define the partitionCH ofH\Gto be the partition intoK-orbits, with respect to the action ofKonH\Gby inverse right translation. The P-collectionC = {CH : H ∈ P} is called the orbit P-scheme associated with the groupK.
This construction indeed yields aP-scheme:
Theorem 2.3. TheP-collectionC in Definition 2.18 is aP-scheme.
Proof. LetK act on each right coset spaceH\Gby inverse right translation. For H, H0 ∈ P withH ⊆H0,g ∈K andx∈H\G, we haveπH,H0(gx) =g(πH,H0(x)) by Lemma 2.2. Therefore if x, x0 ∈ H\G are in the same block of CH (i.e., the sameK-orbit ofH\G), thenπH,H0(x)andπH,H0(x0)are in the same block ofCH0 (i.e., the sameK-orbit ofH0\G). SoC is compatible.
Similarly, forH ∈ P,h ∈G,g ∈Kandx∈H\G, we havecH,h(gx) = g(cH,h(x)) by Lemma 2.2. Therefore ifx, x0 ∈H\Gare in the same block ofCH, thencH,h(x) andcH,h(x0)are in the same block ofCgHg−1. SoC is invariant.
ForH0 ∈ P andy, y0 ∈H0\Gin the same blockB ofCH, chooseg ∈K such that y0 = gy. Asg ∈ K, we havegB =B. ForH ∈ P withH ⊆ H0 andx ∈ H\G, we havex∈B andπH,H0(x) = yiffgx∈gB =B andπH,H0(gx) = g(πH,H0(x)) =
gy=y0. So the mapx7→gxis a one-to-one correspondence betweenB∩π−1H,H0(y) and B ∩ πH,H−1 0(y0), and hence the two sets have the same cardinality. So C is regular.
The connection between Definition 2.17 and Definition 2.18 is given by the following lemma.
Lemma 2.15. For a finite set S, m ∈ N+, and a subgroup K ⊆ Sym(S), let P = Pm be the system of stabilizers of depthm with respect to the natural action of Sym(S) on S, and let C = {CH : H ∈ P} be the orbit P-scheme associated withK. Then the orbitm-scheme associated withK is exactlyΠ(C)as defined in Definition 2.12.
Proof. We may assume m ≤ |S|. Let G be the symmetric groupSym(S)acting naturally on S. Suppose Π(C) = {P1, . . . , Pm} where Pk is a partition of S(k) for k ∈ [m]. By Definition 2.12, each partition Pk is given by Pk = {λ−1x (B) : B ∈ CGx} for some x = (x1, . . . , xk) ∈ S(k), where λx : S(k) → Gx\G is an equivalence between the diagonal action of GonS(k) and the action onGx\G by inverse right translation. It follows thatPkis the partition intoK-orbits with respect to the diagonal action, sinceCGx is the partition into K-orbits with respect to the action by inverse right translation.
Antisymmetry of orbit m-schemes. We prove a simple and exact criterion for antisymmetry of orbitm-schemes.
Lemma 2.16. The orbitm-scheme onSassociated withK ⊆Sym(S)is antisym- metric iff the order ofKis coprime to1,2, . . . , m.
Proof. Let Π = {P1, . . . , Pm} be the orbit m-scheme on S associated with K. Suppose the order ofKis divisible by an integerk satisfying1< k≤m. We may assume thatkis a prime integer. By Cauchy’s theorem (see, e.g., [Lan02]), the group Kcontains an elementgof orderk. The elementg, as a permutation ofS, has at least onek-cycle(x1 x2 · · · xk). Consider the elementx= (x1, . . . , xk)∈S(k), and let Bbe the block ofPkcontainingx. By definition, the elementgx= (gx1, . . . ,gxk) = (x2, . . . , xk, x1)is also inB. On the other hand, leth = (1 2 · · · k)−1 ∈Sym(k). The permutation ckh of S(k) sends x = (x1, . . . , xk) to y = (y1, . . . , yk) defined by yi = xh−1
i for i ∈ [k]. So ckh(x) = (x2, . . . , xk, x1) ∈ B. Therefore Π is not antisymmetric.
Conversely, assumeΠis not antisymmetric. Then for some integerksatisfying1<
k ≤ min{m,|S|},h ∈Sym(k)− {e}, and some elementx = (x1, . . . , xk)∈ S(k) lying in a blockBofPk, we haveckh(x)∈B, i.e.,ckh(x) = gxfor someg ∈K with
respect to the diagonal action ofK onS(k). As the permutationckh ofS(k)sendsx toy= (y1, . . . , yk)defined byyi =xh−1
i fori∈[k], we seegxi =xh−1
i fori∈[k]. Thengpreserves the setT :={x1, . . . , xk}and restricts to a nontrivial permutation g|T ∈ Sym(T) of T. Let e be the order of g|T. Thene is not coprime to some integertwheret≤ |T| ≤m. The order ofKis a multiple of the order ofg, which is a multiple ofe. So the order ofK is not coprime toteither.
Example 2.2. Let S be a finite set satisfying |S| > 1. Let K be a subgroup of Sym(S)generated by a single|S|-cycle so that it acts regularly onS. Denote by ` the least prime factor of|S|. LetΠbe the orbitm-scheme onS associated withK wheremis an integer satisfying1≤m < `. ThenΠis homogeneous sinceK acts transitively onS. The order ofK is|S|, which is coprime to1, . . . , `−1. SoΠis also antisymmetric by Lemma 2.16 and the factm≤`−1.
Upper bound of m for antisymmetric homogeneous m-schemes. Let S be a finite set satisfying|S| > 1, and let` be the least prime factor of|S|. Form ≥ `, the orbit m-schemes on S in Example 2.2 are still homogeneous but no longer antisymmetric. Indeed, an argument of Rónyai [Rón88] shows that form≥`, even generalm-schemes onScannot be both homogeneous and antisymmetric. This was reproduced in [IKS09] and we present it here.
Lemma 2.17([Rón88; IKS09]). LetSbe a finite set satisfying|S|>1, and let`be the least prime factor of|S|. There exists no antisymmetric homogeneousm-scheme onS form≥`.
Proof. Assume to the contrary that such anm-scheme Π = {P1, . . . , Pm} exists.
The groupSym(`)acts onS(`) bygx =ckg(x). By antisymmetry ofΠ, this action induces a semiregular action on the set of blocks in P`. Let B1, . . . , Bk ∈ P` be a complete set of representatives for the Sym(`)-orbits, i.e., each orbit contains exactly oneBi. Then we have
k
X
i=1
|Bi|= |S(`)|
|Sym(`)| = |S|(|S| −1)· · ·(|S| −`+ 1)
`!
Let π be the projection from S(`) to S sending (x1, . . . , x`) to x1. By regularity and homogeneity ofΠ, for eachi∈[k], the cardinality ofBi∩π−1(y)is a constant di ∈N+independent ofy∈S. Then
k
X
i=1
di =
k
X
i=1
|Bi|
|S| = (|S| −1)· · ·(|S| −`+ 1)
`! .
As|S|is a multiple of`, none of the factors|S| −1, . . . ,|S| −`+ 1of the numerator is divisible by the prime number `appeared in the denominator. This contradicts the integrality ofPk
i=1di.
The condition m ≥ ` in Lemma 2.17 is tight, since Example 2.2 shows that anti- symmetric homogeneousm-schemes exist form=`−1.
Rónyai’s result can be extended to P-schemes in the case that P is a system of stabilizers with respect to a transitive group action.
Lemma 2.18. LetGbe a finite group acting transitively on a set S of cardinality n >1. LetP =Pmbe the corresponding system of stabilizers of depthmfor some m ≥ `, where` is the least prime factor ofn. Then for anyx ∈ S, there exists no antisymmetricP-scheme that is homogeneous onGx. In particular,d0(G)< `.
Lemma 2.18 can be easily proven using a technique called the induction of P- schemes, to be discussed in Chapter 6. It allows us to reduce to the caseG= Sym(S). The claim then follows immediately, since by Lemma 2.7, for G = Sym(S), the existence of an antisymmetricP-scheme homogeneous onGximplies the existence of an antisymmetric homogeneousm-scheme onS, which contradicts Lemma 2.17.
For now, we just provide a direct proof.
Proof of Lemma 2.18. Assume to the contrary that C = {CH : H ∈ P} is an antisymmetric P-scheme that is homogeneous on Gx for some x ∈ S. As C is invariant and G acts transitively on S (and hence all one-point stabilizersGx are conjugate inG), we knowC is homogeneous onGxfor allx∈S.
Consider the set S(`) equipped with two actions: the diagonal action of G and the action of Sym(`) permuting the` coordinates. The latter action is defined by
g(x1, . . . , x`) = (xg−1
1, . . . xg−1
`)forg ∈Sym(`)and(x1, . . . , x`)∈S(`). Note that these two actions commute with each other and combine to an action ofG×Sym(`) onS(`). Forz ∈S(`), we havegGz =Ggzfor allg ∈Sym(`)and hence the action ofSym(`)permutes theG-orbits within the(G×Sym(`))-orbit(G×Sym(`))z. Now fixz ∈S(`). We have the bijectionλz :Gz →Gz\Gwhich is an equivalence between the action of G on the G-orbit Gz and the action on Gz\G by inverse right translation. We also have a semiregular action of NG(Gz)/Gz on Gz\G by left translation. This gives a injective group homomorphismφ : NG(Gz)/Gz ,→ Sym(Gz\G), and we denote its image byN. Then|N|=|NG(Gz)/Gz|.
Let H be the subgroup of Sym(`) fixing Gz setwisely, i.e., H = {g ∈ Sym(`) :
gGz = Gz}. The action of H ⊆ Sym(`) on S(`) restricts to an action on Gz and hence we have a group homomorphismH → Sym(Gz). It is injective since elements inGz ⊆ S(`) have distinct coordinates. Now, identifyingGz withGz\G viaλz, we have an action ofH onGz\Gas well, defined by gλz(x) = λz(gx)for x∈Gz. This gives an injective group homomorphismφ0 :H ,→Sym(Gz\G). We claim thatφ0(H)⊆N. To see this, pick anyg ∈H. We havegGze =Gzh0 for someh0 ∈G, or equivalentlygz =h−10 z. Then for anyh∈G, we have
gGzh=g
λz(h−1z)
=λzg
(h−1z)
=λz
h−1
(gz)
=λz((h0h)−1z) =Gzh0h.
In particular, for any h ∈ Gz, we have gGzh = Gzh0h and other other hand
gGzh = gGze =Gzh0. Soh0hh−10 ∈ Gz. Thereforeh0 ∈ NG(Gz). Furthermore, note that h0Gz ∈ NG(Gz)/Gz sends anyGzh ∈ Gz\G toGzh0h = gGzh by left translation. Soφ0(g) =φ(h0Gz)∈N. Thereforeφ0(H)⊆N, as desired.
By antisymmetry, the action of N on Gz\G induces a semiregular action on the set of blocks of CGz, which induces a semiregular action of φ0(H) on the set of blocks ofCGz. Let B1, . . . , Bk ∈ CGz be a complete set of representatives for the φ0(H)-orbits. Then we have
k
X
i=1
|Bi|= |Gz\G|
|φ0(H)| = |Gz|
|H|.
Choose x ∈ S such that Gz ⊆ Gx. By regularity and homogeneity on Gx, for eachi ∈ [k], the cardinality of Bi ∩πG−1z,Gx(y)is a constant di ∈ N+ independent of y ∈ Gx\G, and hence |Bi| is a multiple of |Gx\G| = n. Therefore|Gz| is a multiple ofn· |H|.
By the orbit-stabilizer theorem, the number ofG-orbits contained in(G×Sym(`))z is|Sym(`)|/|H|, and theseG-orbits all have the same cardinality|Gz|. So
|(G×Sym(`))z|= |Sym(`)|
|H| · |Gz|,
which is a multiple ofn · |Sym(`)| = n`! since|Gz| is a multiple ofn· |H|. As this holds for arbitraryz ∈S(`), we know|S(`)|=n(n−1)· · ·(n−`+ 1)is also a multiple ofn`!. But this is not possible sincen−1, . . . , n−`+ 1are not divisible by the prime number`.