Invariance groups of finite functions and orbit equivalence of - - PowerPoint PPT Presentation

Invariance groups of finite functions and

• rbit equivalence of permutation groups

Tam´ as Waldhauser

University of Szeged

NSAC 2013 Novi Sad, 7th June 2013

Joint work with

◮ Eszter Horv´

ath,

◮ Reinhard P¨

• schel,

◮ G´

eza Makay.

Joint work with

◮ Eszter Horv´

ath,

◮ Reinhard P¨

• schel,

◮ G´

eza Makay. We acknowledge helpful discussions with

◮ Erik Friese, ◮ Keith Kearnes, ◮ Erkko Lehtonen, ◮ P3 (P´

eter P´ al P´ alfy),

◮ S´

andor Radeleczki.

Invariance groups

Definition

The invariance group of a function f : kn → m is S (f ) = {σ ∈ Sn | f (x1, . . . , xn) ≡ f (x1σ, . . . , xnσ)} .

Invariance groups

Definition

The invariance group of a function f : kn → m is S (f ) = {σ ∈ Sn | f (x1, . . . , xn) ≡ f (x1σ, . . . , xnσ)} .

Definition

◮ A group G is (k, m)-representable if

there is a function f : kn → m such that S (f ) = G.

Invariance groups

Definition

The invariance group of a function f : kn → m is S (f ) = {σ ∈ Sn | f (x1, . . . , xn) ≡ f (x1σ, . . . , xnσ)} .

Definition

◮ A group G is (k, m)-representable if

there is a function f : kn → m such that S (f ) = G.

◮ A group G is (k, ∞)-representable if

G is (k, m)-representable for some m.

Invariance groups

Definition

The invariance group of a function f : kn → m is S (f ) = {σ ∈ Sn | f (x1, . . . , xn) ≡ f (x1σ, . . . , xnσ)} .

Definition

◮ A group G is (k, m)-representable if

there is a function f : kn → m such that S (f ) = G.

◮ A group G is (k, ∞)-representable if

G is (k, m)-representable for some m. Special cases:

◮ G is (2, 2)-representable iff G is the invariance group of a

Boolean function f : 2n → 2.

Invariance groups

Definition

The invariance group of a function f : kn → m is S (f ) = {σ ∈ Sn | f (x1, . . . , xn) ≡ f (x1σ, . . . , xnσ)} .

Definition

◮ A group G is (k, m)-representable if

there is a function f : kn → m such that S (f ) = G.

◮ A group G is (k, ∞)-representable if

G is (k, m)-representable for some m. Special cases:

◮ G is (2, 2)-representable iff G is the invariance group of a

Boolean function f : 2n → 2.

◮ G is (2, ∞)-representable iff G is the invariance group of a

pseudo-Boolean function f : 2n → m.

Abstract representation

Frucht 1939: Every group is isomorphic to the automorphism group of a graph.

Abstract representation

Frucht 1939: Every group is isomorphic to the automorphism group of a graph.

Corollary

Every group is isomorphic to the invariance group of some Boolean function (i.e., (2, 2)-representable).

Abstract representation

Frucht 1939: Every group is isomorphic to the automorphism group of a graph.

Corollary

Every group is isomorphic to the invariance group of some Boolean function (i.e., (2, 2)-representable).

Proof.

f : 2n → 2 H =

• n, {E ⊆ n | f (χE) = 1}

Abstract representation

Frucht 1939: Every group is isomorphic to the automorphism group of a graph.

Corollary

Every group is isomorphic to the invariance group of some Boolean function (i.e., (2, 2)-representable).

Proof.

f : 2n → 2 H =

• n, {E ⊆ n | f (χE) = 1}
• Example

S

• ∼

= A3

Concrete representation

Example

Suppose that S (f ) = A3 for some f : 23 → m.

Concrete representation

Example

Suppose that S (f ) = A3 for some f : 23 → m. Then f must be constant on the orbits of A3 acting on 23: 000 → a 100, 010, 001 → b 011, 101, 110 → c 111 → d

Concrete representation

Example

Suppose that S (f ) = A3 for some f : 23 → m. Then f must be constant on the orbits of A3 acting on 23: 000 → a 100, 010, 001 → b 011, 101, 110 → c 111 → d However, such a function is totally symmetric, i.e., S (f ) = S3.

Concrete representation

Example

Suppose that S (f ) = A3 for some f : 23 → m. Then f must be constant on the orbits of A3 acting on 23: 000 → a 100, 010, 001 → b 011, 101, 110 → c 111 → d However, such a function is totally symmetric, i.e., S (f ) = S3. Thus A3 is not (2, ∞)-representable.

Concrete representation

Example

Suppose that S (f ) = A3 for some f : 23 → m. Then f must be constant on the orbits of A3 acting on 23: 000 → a 100, 010, 001 → b 011, 101, 110 → c 111 → d However, such a function is totally symmetric, i.e., S (f ) = S3. Thus A3 is not (2, ∞)-representable. Let g : 33 → 2 such that g (0, 1, 2) = g (1, 2, 0) = g (2, 1, 0) = 1 and g = 0 everywhere else. Then S (g) = A3, thus A3 is (3, 2)-representable.

Ein Kleines Problem

Clote, Kranakis 1991: If G is (2, ∞)-representable, then G is (2, 2)-representable.

Ein Kleines Problem

Clote, Kranakis 1991: If G is (2, ∞)-representable, then G is (2, 2)-representable. Kisielewicz 1998: False!

Ein Kleines Problem

Clote, Kranakis 1991: If G is (2, ∞)-representable, then G is (2, 2)-representable. Kisielewicz 1998: False! The Klein four-group V = {id, (12) (34) , (13) (24) , (14) (23)} ≤ S4 is a counterexample;

Ein Kleines Problem

Clote, Kranakis 1991: If G is (2, ∞)-representable, then G is (2, 2)-representable. Kisielewicz 1998: False! The Klein four-group V = {id, (12) (34) , (13) (24) , (14) (23)} ≤ S4 is a counterexample; moreover, it is the only counterexample that

• ne could “easily” find.

Ein Kleines Problem

Clote, Kranakis 1991: If G is (2, ∞)-representable, then G is (2, 2)-representable. Kisielewicz 1998: False! The Klein four-group V = {id, (12) (34) , (13) (24) , (14) (23)} ≤ S4 is a counterexample; moreover, it is the only counterexample that

• ne could “easily” find.

V = S

Ein Kleines Problem

Clote, Kranakis 1991: If G is (2, ∞)-representable, then G is (2, 2)-representable. Kisielewicz 1998: False! The Klein four-group V = {id, (12) (34) , (13) (24) , (14) (23)} ≤ S4 is a counterexample; moreover, it is the only counterexample that

• ne could “easily” find.

V = S

• =

⇒ V is (2, 3)-representable but not (2, 2)-representable.

Ein Kleines Problem

Clote, Kranakis 1991: If G is (2, ∞)-representable, then G is (2, 2)-representable. Kisielewicz 1998: False! The Klein four-group V = {id, (12) (34) , (13) (24) , (14) (23)} ≤ S4 is a counterexample; moreover, it is the only counterexample that

• ne could “easily” find.

V = S

• =

⇒ V is (2, 3)-representable but not (2, 2)-representable. Dalla Volta, Siemons 2012: There are infinitely many groups that are (2, ∞)-representable but not (2, 2)-representable. (?)

Ein Kleines Problem

Clote, Kranakis 1991: If G is (2, ∞)-representable, then G is (2, 2)-representable. Kisielewicz 1998: False! The Klein four-group V = {id, (12) (34) , (13) (24) , (14) (23)} ≤ S4 is a counterexample; moreover, it is the only counterexample that

• ne could “easily” find.

V = S

• = S
• ∩

S

• =

⇒ V is (2, 3)-representable but not (2, 2)-representable. Dalla Volta, Siemons 2012: There are infinitely many groups that are (2, ∞)-representable but not (2, 2)-representable. (?)

Orbit closure

Clote, Kranakis 1991: The following are equivalent for any group G ≤ Sn: (i) G is the invariance group of a pseudo-Boolean function (i.e., G is (2, ∞)-representable). (ii) G is the intersection of invariance groups of Boolean functions (i.e., (2, 2)-representable groups).

Orbit closure

Clote, Kranakis 1991: The following are equivalent for any group G ≤ Sn: (i) G is the invariance group of a pseudo-Boolean function (i.e., G is (2, ∞)-representable). (ii) G is the intersection of invariance groups of Boolean functions (i.e., (2, 2)-representable groups). (iii) G is orbit closed.

Orbit closure

Clote, Kranakis 1991: The following are equivalent for any group G ≤ Sn: (i) G is the invariance group of a pseudo-Boolean function (i.e., G is (2, ∞)-representable). (ii) G is the intersection of invariance groups of Boolean functions (i.e., (2, 2)-representable groups). (iii) G is orbit closed. Two subgroups of Sn are orbit equivalent if they have the same

• rbits on P (n)

Orbit closure

Clote, Kranakis 1991: The following are equivalent for any group G ≤ Sn: (i) G is the invariance group of a pseudo-Boolean function (i.e., G is (2, ∞)-representable). (ii) G is the intersection of invariance groups of Boolean functions (i.e., (2, 2)-representable groups). (iii) G is orbit closed. Two subgroups of Sn are orbit equivalent if they have the same

• rbits on P (n) 2n.

Orbit closure

Clote, Kranakis 1991: The following are equivalent for any group G ≤ Sn: (i) G is the invariance group of a pseudo-Boolean function (i.e., G is (2, ∞)-representable). (ii) G is the intersection of invariance groups of Boolean functions (i.e., (2, 2)-representable groups). (iii) G is orbit closed. Two subgroups of Sn are orbit equivalent if they have the same

• rbits on P (n) 2n.

The orbit closure of G is the greatest element of its

• rbit equivalence class.

Primitive groups

Inglis; Cameron, Neumann, Saxl; Siemons, Wagner 1984–85: Almost all primitive groups are orbit closed.

Primitive groups

Inglis; Cameron, Neumann, Saxl; Siemons, Wagner 1984–85: Almost all primitive groups are orbit closed. Seress 1997: All primitive subgroups of Sn are orbit closed except for An and C5, AGL (1, 5), PGL (2, 5), AGL (1, 8), AΓL(1, 8), AGL (1, 9), ASL (2, 3), PSL (2, 8), PΓL(2, 8) and PGL (2, 9).

Primitive groups

Inglis; Cameron, Neumann, Saxl; Siemons, Wagner 1984–85: Almost all primitive groups are orbit closed. Seress 1997: All primitive subgroups of Sn are orbit closed except for An and C5, AGL (1, 5), PGL (2, 5), AGL (1, 8), AΓL(1, 8), AGL (1, 9), ASL (2, 3), PSL (2, 8), PΓL(2, 8) and PGL (2, 9).

Theorem

All primitive groups are (3, ∞)-representable except for the alternating groups.

A Galois connection

For a = (a1, . . . , an) ∈ kn and σ ∈ Sn, let aσ = (a1σ, . . . , anσ) .

n: number of variables, k: size of domain

A Galois connection

For a = (a1, . . . , an) ∈ kn and σ ∈ Sn, let aσ = (a1σ, . . . , anσ) . If f : kn → k and σ ∈ Sn, then we write σ ⊢ f : ⇐ ⇒ f (aσ) = f (a) for all a ∈ kn.

n: number of variables, k: size of domain

A Galois connection

For a = (a1, . . . , an) ∈ kn and σ ∈ Sn, let aσ = (a1σ, . . . , anσ) . If f : kn → k and σ ∈ Sn, then we write σ ⊢ f : ⇐ ⇒ f (aσ) = f (a) for all a ∈ kn. Let O(n)

k

= {f | f : kn → k}, and for F ⊆ O(n)

k

and G ⊆ Sn define F ⊢ := {σ ∈ Sn | ∀f ∈ F : σ ⊢ f }, F (k) := (F ⊢)⊢, G ⊢ := {f ∈ O(n)

k

| ∀σ ∈ G : σ ⊢ f }, G (k) := (G ⊢)⊢.

n: number of variables, k: size of domain

A Galois connection

For a = (a1, . . . , an) ∈ kn and σ ∈ Sn, let aσ = (a1σ, . . . , anσ) . If f : kn → k and σ ∈ Sn, then we write σ ⊢ f : ⇐ ⇒ f (aσ) = f (a) for all a ∈ kn. Let O(n)

k

= {f | f : kn → k}, and for F ⊆ O(n)

k

and G ⊆ Sn define F ⊢ := {σ ∈ Sn | ∀f ∈ F : σ ⊢ f }, F (k) := (F ⊢)⊢, G ⊢ := {f ∈ O(n)

k

| ∀σ ∈ G : σ ⊢ f }, G (k) := (G ⊢)⊢. For G ≤ Sn, we call G (k) the Galois closure of G over k.

n: number of variables, k: size of domain

Galois closed groups as invariance groups

n: number of variables, k: size of domain

Galois closed groups as invariance groups

Fact

The following are equivalent for any group G ≤ Sn: (i) G is Galois closed over k. (ii) G is (k, ∞)-representable.

n: number of variables, k: size of domain

Galois closed groups as invariance groups

Fact

The following are equivalent for any group G ≤ Sn: (i) G is Galois closed over k. (ii) G is (k, ∞)-representable. (iii) G is the invariance group of a function f : kn → ∞.

n: number of variables, k: size of domain

Galois closed groups as invariance groups

Fact

The following are equivalent for any group G ≤ Sn: (i) G is Galois closed over k. (ii) G is (k, ∞)-representable. (iii) G is the invariance group of a function f : kn → ∞. (iv) G is the intersection of invariance groups of functions kn → 2. (v) G is the intersection of invariance groups of functions kn → k.

n: number of variables, k: size of domain

Galois closed groups as invariance groups

Fact

The following are equivalent for any group G ≤ Sn: (i) G is Galois closed over k. (ii) G is (k, ∞)-representable. (iii) G is the invariance group of a function f : kn → ∞. (iv) G is the intersection of invariance groups of functions kn → 2. (v) G is the intersection of invariance groups of functions kn → k. (vi) G is orbit closed with respect to the action of Sn on kn.

n: number of variables, k: size of domain

Orbits and closures

For a = (a1, . . . , an) ∈ kn and G ≤ Sn, define aG = {aσ | σ ∈ G} , Orb(k) (G) =

• aG | a ∈ kn

.

n: number of variables, k: size of domain

Orbits and closures

For a = (a1, . . . , an) ∈ kn and G ≤ Sn, define aG = {aσ | σ ∈ G} , Orb(k) (G) =

• aG | a ∈ kn

. For all G, H ≤ Sn we have G (k) = H(k) ⇐ ⇒ G ⊢ = H⊢

n: number of variables, k: size of domain

Orbits and closures

For a = (a1, . . . , an) ∈ kn and G ≤ Sn, define aG = {aσ | σ ∈ G} , Orb(k) (G) =

• aG | a ∈ kn

. For all G, H ≤ Sn we have G (k) = H(k) ⇐ ⇒ G ⊢ = H⊢ ⇐ ⇒ Orb(k) (G) = Orb(k) (H) ;

n: number of variables, k: size of domain

Orbits and closures

For a = (a1, . . . , an) ∈ kn and G ≤ Sn, define aG = {aσ | σ ∈ G} , Orb(k) (G) =

• aG | a ∈ kn

. For all G, H ≤ Sn we have G (k) = H(k) ⇐ ⇒ G ⊢ = H⊢ ⇐ ⇒ Orb(k) (G) = Orb(k) (H) ; G (k) =

• σ ∈ Sn | ∀a ∈ kn : aσ ∈ aG

.

n: number of variables, k: size of domain

Orbits and closures

For a = (a1, . . . , an) ∈ kn and G ≤ Sn, define aG = {aσ | σ ∈ G} , Orb(k) (G) =

• aG | a ∈ kn

. For all G, H ≤ Sn we have G (k) = H(k) ⇐ ⇒ G ⊢ = H⊢ ⇐ ⇒ Orb(k) (G) = Orb(k) (H) ; G (k) =

• σ ∈ Sn | ∀a ∈ kn : aσ ∈ aG

. The case k = 2 corresponds to orbit equivalence and orbit closure.

n: number of variables, k: size of domain

Orbits and closures

For a = (a1, . . . , an) ∈ kn and G ≤ Sn, define aG = {aσ | σ ∈ G} , Orb(k) (G) =

• aG | a ∈ kn

. For all G, H ≤ Sn we have G (k) = H(k) ⇐ ⇒ G ⊢ = H⊢ ⇐ ⇒ Orb(k) (G) = Orb(k) (H) ; G (k) =

• σ ∈ Sn | ∀a ∈ kn : aσ ∈ aG

. The case k = 2 corresponds to orbit equivalence and orbit closure.

Proposition

For all G ≤ Sn we have G (2) ≥ G (3) ≥ · · ·≥ G (n) = · · ·= G.

n: number of variables, k: size of domain

A formula for the closure

Proposition

For every G ≤ Sn and k ≥ 2, we have G (k) =

• a∈kn

Stab (a) · G.

n: number of variables, k: size of domain

A formula for the closure

Proposition

For every G ≤ Sn and k ≥ 2, we have G (k) =

• a∈kn

Stab (a) · G.

Proof.

G (k) =

• σ ∈ Sn | ∀a ∈ kn : aσ ∈ aG

n: number of variables, k: size of domain

A formula for the closure

Proposition

For every G ≤ Sn and k ≥ 2, we have G (k) =

• a∈kn

Stab (a) · G.

Proof.

G (k) =

• σ ∈ Sn | ∀a ∈ kn : aσ ∈ aG

= {σ ∈ Sn | ∀a ∈ kn ∃π ∈ G : aσ = aπ}

n: number of variables, k: size of domain

A formula for the closure

Proposition

For every G ≤ Sn and k ≥ 2, we have G (k) =

• a∈kn

Stab (a) · G.

Proof.

G (k) =

• σ ∈ Sn | ∀a ∈ kn : aσ ∈ aG

= {σ ∈ Sn | ∀a ∈ kn ∃π ∈ G : aσ = aπ} =

• σ ∈ Sn | ∀a ∈ kn ∃π ∈ G : σπ−1 ∈ Stab (a)
• n: number of variables,

k: size of domain

A formula for the closure

Proposition

For every G ≤ Sn and k ≥ 2, we have G (k) =

• a∈kn

Stab (a) · G.

Proof.

G (k) =

• σ ∈ Sn | ∀a ∈ kn : aσ ∈ aG

= {σ ∈ Sn | ∀a ∈ kn ∃π ∈ G : aσ = aπ} =

• σ ∈ Sn | ∀a ∈ kn ∃π ∈ G : σπ−1 ∈ Stab (a)
• = {σ ∈ Sn | ∀a ∈ kn : σ ∈ Stab (a) · G}

n: number of variables, k: size of domain

A formula for the closure

Proposition

For every G ≤ Sn and k ≥ 2, we have G (k) =

• a∈kn

Stab (a) · G.

Proof.

G (k) =

• σ ∈ Sn | ∀a ∈ kn : aσ ∈ aG

= {σ ∈ Sn | ∀a ∈ kn ∃π ∈ G : aσ = aπ} =

• σ ∈ Sn | ∀a ∈ kn ∃π ∈ G : σπ−1 ∈ Stab (a)
• = {σ ∈ Sn | ∀a ∈ kn : σ ∈ Stab (a) · G}

=

• a∈kn

Stab (a) · G.

n: number of variables, k: size of domain

The case k = n − 1

Theorem

If k = n − 1 ≥ 2, then all subgroups of Sn except An are Galois closed over k.

n: number of variables, k: size of domain

The case k = n − 1

Theorem

If k = n − 1 ≥ 2, then all subgroups of Sn except An are Galois closed over k.

Definition (Clote, Kranakis 1991)

A group G ≤ Sn is weakly representable, if G is (k, ∞)-representable for some k < n.

n: number of variables, k: size of domain

The case k = n − 1

Theorem

If k = n − 1 ≥ 2, then all subgroups of Sn except An are Galois closed over k.

Definition (Clote, Kranakis 1991)

A group G ≤ Sn is weakly representable, if G is (k, ∞)-representable for some k < n.

Corollary

All subgroups of G ≤ Sn except for An are weakly representable.

n: number of variables, k: size of domain

The case k = n − 1

Theorem

If k = n − 1 ≥ 2, then all subgroups of Sn except An are Galois closed over k.

Definition (Clote, Kranakis 1991)

A group G ≤ Sn is weakly representable, if G is (k, ∞)-representable for some k < n.

Corollary

All subgroups of G ≤ Sn except for An are weakly representable.

Proof.

G ≤ Sn is weakly representable ⇐ ⇒ ∃k < n : G (k) = G

n: number of variables, k: size of domain

The case k = n − 1

Theorem

If k = n − 1 ≥ 2, then all subgroups of Sn except An are Galois closed over k.

Definition (Clote, Kranakis 1991)

A group G ≤ Sn is weakly representable, if G is (k, ∞)-representable for some k < n.

Corollary

All subgroups of G ≤ Sn except for An are weakly representable.

Proof.

G ≤ Sn is weakly representable ⇐ ⇒ ∃k < n : G (k) = G ⇐ ⇒ G (n−1) = G

n: number of variables, k: size of domain

The case k = n − 1

Theorem

If k = n − 1 ≥ 2, then all subgroups of Sn except An are Galois closed over k.

Definition (Clote, Kranakis 1991)

A group G ≤ Sn is weakly representable, if G is (k, ∞)-representable for some k < n.

Corollary

All subgroups of G ≤ Sn except for An are weakly representable.

Proof.

G ≤ Sn is weakly representable ⇐ ⇒ ∃k < n : G (k) = G ⇐ ⇒ G (n−1) = G ⇐ ⇒ G = An

n: number of variables, k: size of domain

The case k = n − 2

Theorem

If k = n − 2 ≥ 2, then the Galois closures of subgroups of Sn are:

◮ An (k) = Sn;

n: number of variables, k: size of domain

The case k = n − 2

Theorem

If k = n − 2 ≥ 2, then the Galois closures of subgroups of Sn are:

◮ An (k) = Sn; ◮ An−1 (k) = Sn−1;

n: number of variables, k: size of domain

The case k = n − 2

Theorem

If k = n − 2 ≥ 2, then the Galois closures of subgroups of Sn are:

◮ An (k) = Sn; ◮ An−1 (k) = Sn−1; ◮ C4 (k) = D4 (for n = 4);

n: number of variables, k: size of domain

The case k = n − 2

Theorem

If k = n − 2 ≥ 2, then the Galois closures of subgroups of Sn are:

◮ An (k) = Sn; ◮ An−1 (k) = Sn−1; ◮ C4 (k) = D4 (for n = 4); ◮ all other subgroups of Sn are closed.

n: number of variables, k: size of domain

The case k = n − d

Theorem

Let n > max

• 2d, d2 + d
• and G ≤ Sn. Then G is not Galois

closed over k if and only if

• 1. G ≤sd AL × ∆ or
• 2. G <sd SL × ∆,

where n = L ˙ ∪ D with |L| > d, |D| < d and ∆ ≤ SD. The closure of these groups is G (k) = SL × ∆.

n: number of variables, k: size of domain

The case k = n − d

Theorem

Let n > max

• 2d, d2 + d
• and G ≤ Sn. Then G is not Galois

closed over k if and only if

• 1. G ≤sd AL × ∆ or
• 2. G <sd SL × ∆,

where n = L ˙ ∪ D with |L| > d, |D| < d and ∆ ≤ SD. The closure of these groups is G (k) = SL × ∆.

Remark

Using the simplicity of alternating groups, one can show that these subdirect products are of the following form:

• 1. G = AL × ∆;

n: number of variables, k: size of domain

The case k = n − d

Theorem

Let n > max

• 2d, d2 + d
• and G ≤ Sn. Then G is not Galois

closed over k if and only if

• 1. G ≤sd AL × ∆ or
• 2. G <sd SL × ∆,

where n = L ˙ ∪ D with |L| > d, |D| < d and ∆ ≤ SD. The closure of these groups is G (k) = SL × ∆.

Remark

Using the simplicity of alternating groups, one can show that these subdirect products are of the following form:

• 1. G = AL × ∆;
• 2. G = (AL × ∆0) ∪

(SL \ AL) × (∆ \ ∆0)

• ,

where ∆0 ≤ ∆ is a subgroup of index 2.

n: number of variables, k: size of domain

Interesting subgroups of S4, S5 and S6

Interesting subgroups of S4, S5 and S6

G ≤ Sn G (2) G (3) G (4) C4 D4 C4 C4

Interesting subgroups of S4, S5 and S6

G ≤ Sn G (2) G (3) G (4) C4 D4 C4 C4 C5 D5 C5 C5 AGL (1, 5) S5 AGL (1, 5) AGL (1, 5)

Interesting subgroups of S4, S5 and S6

G ≤ Sn G (2) G (3) G (4) C4 D4 C4 C4 C5 D5 C5 C5 AGL (1, 5) S5 AGL (1, 5) AGL (1, 5) C4 × S2 D4 × S2 C4 × S2 C4 × S2 D4 ×sd S2 D4 × S2 D4 ×sd S2 D4 ×sd S2 A3 ≀ A2 S3 ≀ S2 A3 ≀ A2 A3 ≀ A2 S3 ≀sd S2 S3 ≀ S2 S3 ≀sd S2 S3 ≀sd S2 (S3 ≀ S2) ∩ A6 S3 ≀ S2 S3 ≀ S2 (S3 ≀ S2) ∩ A6 PGL (2, 5) S6 PGL (2, 5) PGL (2, 5) Rot () Sym () Rot () Rot ()

References

P.J. Cameron, P.M. Neumann, J. Saxl, On groups with no regular orbits on the set of subsets, Arch. Math. 43 (1984), 295–296.

• P. Clote, E. Kranakis, Boolean functions, invariance groups, and parallel

complexity, SIAM J. Comput. 20 (1991), 553–590.

• F. Dalla Volta, J. Siemons, Orbit equivalence and permutation groups defined

by unordered relations, J. Algebr. Comb. 35 (2012), 547–564.

• E. K. Horv´

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