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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Invariance groups of functions, orbit equivalence and group actions

Reinhard P¨

• schel

Institut f¨ ur Algebra Technische Universit¨ at Dresden

Workshop on Algebraic Graph Theory Plzeˇ n, October 3, 2016

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (1/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Joint work with

• Eszter Horv´

ath,

• G´

eza Makay,

• Tamas Waldhauser*.

2012: Invariance groups of finite functions and orbit equivalence of permutation groups, arXiv:1210.1015. *invited talk at AAA85 (2013, Luxembourg)

We acknowledge helpful discussions with

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

eter P´ al P´ alfy),

• S´

andor Radeleczki,

• Sven Reichard.
• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (2/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Outline

Invariance groups The closure under a general point of view: group actions and a Galois connection Characterizations of the closure Concrete results Another closure and Galois connection

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (3/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

The orginal problem: Symmetry groups of Boolean functions

The symmetry group or invariance group of a Boolean function f : 2n → 2 (2 := {0, 1}) is f ⊢ := {σ ∈ Sym(n) | ∀x1, . . . , xn ∈ 2 : f (x1σ, . . . , xnσ) = f (x1, . . . , xn)}. Notation: σ : i → iσ (action of σ ∈ Sym(n) on i ∈ {1, . . . , n}) xσ = (x1, . . . , xn)σ := (x1σ, . . . , xnσ) (action of σ on x ∈ 2n) Thus f ⊢ = {σ ∈ Sym(n) | ∀x ∈ 2n : f (xσ) = f (x)} Problem: Which groups are such symmetry groups?

• A. Kisielewicz, Symmetry groups of Boolean functions and

constructions of permutation groups. J. of Algebra 1998, (1998), 379–403.

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (4/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Examples and Non-Examples

Problem: Which groups are symmetry groups of Boolean functions? What means are? Does it mean isomorphic? Then we have Every group G ≤ Sym(n) is isomorphic to the symmetry group of a Boolean function.

Proof.

recall the result of Frucht (1939) Every group is isomorphic to the automorphism group of a graph. and notice the 1-1-correspondence: f : 2n → 2 H = (n, {E ∈ P(n) | f (χE) = 1}) (hypergraph) Example: f ⊢ = Aut

• ∼

=A3

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (5/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Examples and Non-Examples

Problem: Which groups are symmetry groups of Boolean functions? What means are? Does it mean equality? Then there are groups which are not symmetry groups of a Boolean function, e.g., for the alternating group An ≤ Sym(n) there is no f with f ⊢=An. Suppose that f ⊢ = A3 for some f : 2n → 2. 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., f ⊢ = S3 = A3.

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Generalization

Instead of f : 2n → 2 consider functions f : kn → m invariance group f ⊢ = {σ ∈ Sym(n) | ∀x1, . . . , xn ∈ k : f (x1σ, . . . , xnσ) = f (x1, . . . , xn)}

Definition

• A group G ≤ Sym(n) is called (k, m)-representable if there

exists a function f : kn → m such that G = f ⊢.

• A group G ≤ Sym(n) is called (k, ∞)-representable if it is

(k, m)-representable for some m ∈ N+. Thus G is (2, 2)-representable iff it is the invariance group of a Boolean function.

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Pseudo-Boolean functions

A group G ≤ Sym(n) is (2, m)-representable iff it is the invariance group of a pseudo-Boolean function f : 2n → m. ( hypergraph on n vertices with colored edges (m − 1 colors)) Clote, Kranakis 1991: If G is the invariance group of a pseudo-Boolean function, then G is the invariance group of a Boolean function. Kisielewicz 1998: False! The Klein four-group V is a counterexample; moreover, it is the only counterexample that one could “easily” find.

Conjecture (Kisielewicz)

There are infinitely many groups that are (2, ∞)-representable but not (2, 2)-representable.

An equivalent result is given in Dalla Volta, Siemons (2012), Corollary 5.3, however its proof contains a gap.

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Example: Klein’s four-group

The Klein four-group V = {id, (12)(34), (13)(24), (14)(23)} ≤ Sym(4) is not (2, 2)-representable but it is the intersection of the invariance groups of two Boolean functions: V = Aut

• = Aut
• ∩

Aut

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (9/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Invariance groups and orbit closed groups

Clote, Kranakis 1991: The following are equivalent for any group G ≤ Sym(n): (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., G = F ⊢ for a set F of Boolean functions), (iii) G is orbit closed.

Definition

Two subgroups of Sym(n) are orbit equivalent if they have the same orbits on P(n). The orbit closure of G is the greatest element of its orbit equivalence class. Note: action (2n, Sym(n)) action (P(n), Sym(n)). a := (a1, . . . , an) ∈ 2n ← → a := {i ∈ {1, . . . , n} | ai = 1} ∈ P(n)

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Example: primitive groups

Inglis; Cameron, Neumann, Saxl; Siemons,Wagner 1984-85: Almost all primitive groups are orbit closed. Seress 1997: All primitive subgroups of Sym(n) are orbit closed (equivalently, (2, ∞)-representable) except for An and C5, AGL(1,5), PGL(2,5), AGL(1,8), AGL(1,8), AGL(1,9), ASL(2,3), PSL(2,8), PGL(2,8) and PGL(2,9). Horv´ ath, Makay, P¨

• schel, Waldhauser 2014:

Every primitive permutation group except for An (n ≥ 4) is (3, ∞)-representable.

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (11/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Outline

Invariance groups The closure under a general point of view: group actions and a Galois connection Characterizations of the closure Concrete results Another closure and Galois connection

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Example: functions f : kn → m

Recall: invariance group for f : kn → m f ⊢ = {σ ∈ Sym(n) | ∀x1, . . . , xn ∈ k : f (x1σ, . . . , xnσ) = f (x1, . . . , xn)} f (xσ) = f (x) for action σ ∈ Sym(n) on x ∈ kn The following are equivalent for any group G ≤ Sym(n): (i) G is (k, ∞)-representable. (ii) G is the invariance group of a function f : kn → N. (iii) G is the intersection of invariance groups of functions kn → 2. (iv) G is the intersection of invariance groups of functions kn → k. (v) G is Galois closed with respect to the Galois connection induced by ⊢.

here ⊢ ⊆ O(n)

k

× Sym(n) is a binary relation between n-ary functions (on k = {0, . . . , k − 1}) and permutations from Sym(n).

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (13/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

General setting: Galois connections

The Galois connection induced by a binary relation R ⊆ G × M is given by the pair of mappings ϕ : P(G) → P(M) : X → X R := {m ∈ M | ∀g ∈ X : gRm} ψ : P(M) → P(G) : Y → Y R := {g ∈ G | ∀m ∈ Y : gRm} Galois closures X = (X R)R, Y = (Y R)R A Galois connection (ϕ, ψ) is characterizable by the property ∀ X ⊆ G, Y ⊆ M : Y ⊆ ϕ(X) ⇐ ⇒ ψ(Y ) ⊇ X

In Formal Concept Analysis (FCA)(Ganter/Wille): G: objects (Gegenst¨ ande), M: attributes (Merkmale), gRm: object g has attribute m

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

General setting: group actions

Γ = (Γ, ·, ε) group (with identity element ε) e.g. Γ = Sym(n) (A, Γ) group action (Γ acts on a set A): A := kn mapping A × Γ → A : (a, σ) → aσ (x1, . . . , xn)σ := (x1σ, . . . , xnσ) such that xε = x (xσ)τ = xστ for all x ∈ A and σ, τ ∈ Γ.

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (15/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

The Galois connection induced by ⊢

(A, Γ) group action, K arbitrary set (e.g., A = 2n, K = 2, Γ = Sym(n)) ⊢ relation between group elements σ ∈ Γ and functions f : A → K

Definition

σ ⊢ f : ⇐ ⇒ ∀x ∈ A : f (xσ) = f (x). Then f ∈ K A is called an invariant for σ ∈ Γ and σ is called a symmetry of f . A

action

− − − − →

x→xσ

A

• f

 

• A

f

− − − − → K Clearly, σ ⊢ f if and only if σ−1 ⊢ f . Corresponding Galois connection (let F ⊆ K A and G ⊆ Γ) F ⊢ := {σ ∈ Γ | ∀f ∈ F : σ ⊢ f }, G ⊢ := {f ∈ K A | ∀σ ∈ G : σ ⊢ f }, Galois closures: F := (F ⊢)⊢, G := (G ⊢)⊢ .

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (16/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Outline

Invariance groups The closure under a general point of view: group actions and a Galois connection Characterizations of the closure Concrete results Another closure and Galois connection

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (17/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

The problem

easy to check: Galois closures G = (G ⊢)⊢ are always subgroups of Γ.

Problem

Given a group action (A, Γ), characterize the Galois closed subgroups G = G.

Such Galois closed subgroups are of the form G = F ⊢ (invariance group of functions, special case G = f ⊢ := {f }⊢)

some notation: For a subgroup G ≤ Γ let OrbA G := {aG | a ∈ A} (where aG := {aσ | σ ∈ G}) (set of all orbits of G (under the group action)). For a ∈ A and B ⊆ A let Γa := {σ ∈ Γ | aσ = a} (stabilizer of a). ΓB := {σ ∈ Γ | Bσ = B} (set-stabilizer of set B).

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Characterizing G

From b, b′ ∈ B ∈ Orb(G) ⇐ ⇒ ∃σ ∈ G : b′ = bσ directly follows: Observation: The following conditions are equivalent (for G ≤ Γ, f ∈ K A): (i) f ∈ G ⊢, (ii) f is constant on each B ∈ Orb G, (iii) In case of K = 2 (w.l.o.g): The support B := supp f := {a ∈ A | f (a) = 1} ⊆ A of f is invariant under the action (A, G), i.e., ∀σ ∈ G : Bσ = B.

Corollary

The Galois closure G is the largest subgroup among all subgroups

• f Γ with the same orbits (on A) as G:

G = {σ ∈ Γ | ∀a ∈ A : aσ ∈ aG} = {σ ∈ Γ | ∀B ∈ Orb(G) : Bσ = B} =

• B∈Orb(G)

ΓB.

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (19/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Characterizing G, continued

Proposition

Let (A, Γ) be a group action and G ≤ Γ. Then we have:1 G =

• a∈A

Γa · G. (*)

Proof.

G = {σ ∈ Γ | ∀a ∈ A : aσ ∈ aG} = {σ ∈ Γ | ∀a ∈ A ∃π ∈ G : aσ = aπ} = {σ ∈ Γ | ∀a ∈ A ∃π ∈ G : σπ−1 ∈ Γa} = {σ ∈ Γ | ∀a ∈ A : σ ∈ Γa · G} =

• a∈A

Γa · G.

1For the action (A, Γ) = (2n, Sym(n)), (*) was formulated and proved by E.

Horvath, K. Kearnes.

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (20/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Further research

study and find the Galois closed groups for different concrete group actions (A, Γ). For example:

• For the action (2n, Sym(n)) or (kn, Sym(n)): next section
• For the action (2n, GLn(2))

(equivalently, (P(n), GLn(2))):

• W. Xiao: Linear symmetries of Boolean functions. (2005)

some preliminary results: E. Horv´ ath, R. P¨

• schel, S. Reichard

(2014 – ...)

Γ := GL(n, 2) (general linear group) acting on A := 2n: action of a regular (n × n)-matrix M ∈ GL(n, 2) (over 2-element field GF(2)) on x = (x1, . . . , xn)⊤ (considered as column vector) by matrix multiplication: xM := M x, (all computations in GF(2)).

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (21/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Outline

Invariance groups The closure under a general point of view: group actions and a Galois connection Characterizations of the closure Concrete results Another closure and Galois connection

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (22/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Back to the actions (kn, Sym(n))

action of Sym(n) on kn (n ≥ 1 fixed) Galois closure of G ≤ Sym(n), notation now: G

(k) := (G ⊢)⊢

(Galois closure over k) recall: G

(k) is the greatest group with the same orbits on kn as the

action of G on kn, and G Galois closed over k ⇐ ⇒ G is intersection of invariance groups

• f functions f : kn → k ⇐

⇒ G is (k, ∞)-representable k = 2: orbit closure (action on P(n))

Proposition (Refinement of closures)

For all G ≤ Sym(n) we have G

(2) ≥ G (3) ≥ · · · ≥ G (n−1) ≥ G (n) = G (n+1) = · · · = G.

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

The case k = n − 1

Sn := Sym(n)

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

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

The case k = n − 2

Theorem

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

• An

(k) = Sn,

• An−1

(k) = Sn−1,

• C4

(k) = D4 (for n = 4),

• all other subgroups of Sn are closed.
• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

The case k = n − d

Theorem

Let n > max

• 2d, d2 + d
• and G ≤ Sn.(e.g., d = 3 =

⇒ n > 12) Then G is not Galois closed over k if and only if

• 1. G = AB × L or
• 2. G <sd SB × L,

where B ⊆ n is such that D := n \ B has less than d elements, and L is an arbitrary permutation group on D. The closure of these groups is G

(k) = SB × L.

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

• 2. G = (AB × L0) ∪
• (SB \ AB) × (L \ L0)
• , where L0 ≤ L is a

subgroup of index 2.

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Some 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 ()

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

What could be done further ... ?

• Increase d (case k = n − d): describe (non-)closed subgroups
• f Sn.
• Decrease the bound max
• 2d, d2 + d
• n n.
• Determine the closures of your favourite groups.
• Give a “human” proof for: G primitive =

⇒ G

(3) = G.

• Find transitive groups G with G

(k) = G for arbitrarily large

values of k.

• Investigate (k, m)-representable groups.
• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Outline

Invariance groups The closure under a general point of view: group actions and a Galois connection Characterizations of the closure Concrete results Another closure and Galois connection

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (29/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

“Classical” Galois connection between permutations and relations

G = Sn, M = ∞

m=1 P(Am) (m-ary relations) and R = ⊲

σ ⊲ B

(permutation σ ∈ Sym(A) preserves a relation B ⊆ Am) (relation B is invariant for σ) (σ is an automorphism of B) if r := (r1, . . . , rm) ∈ B implies rσ := (rσ

1 , . . . , rσ m) ∈ B

Galois closures (for finite A): (F ⊲)⊲ = Aut Inv F (permutation groups) (Q⊲)⊲ = Inv Aut Q (Krasner algebras)

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Which relations characterize G

(k)?

Every permutation group G ≤ Sn is the automorphism group of relations. Which relations characterize the Galois closures G

(k)?

Answer for k = 2 (invariance groups of Boolean functions):

Proposition

The Galois closure G

(2) (= orbit closure) of G ⊆ Sn is given by

G

(2) = Aut Inv(n) sym G

(= Aut Invsym G). Invsym G = totally symmetric relations which are invariant for G.

2.10.2016: Hint by M. Klin: strong connection to results of D. Betten: Geometrische Permutationsgruppen (1977), and (w.r.t. Wielandt’s closure) connections to results of L.A. Kaluˇ znin, M.H. Klin, On some numerical invariants of permutation groups (1976)

Open problem: similar characterization for G

(k) (k > 2)

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (31/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Wielandt’s closure

refining the closure: Which permutation groups are characterizable by k-ary relations? Wielandt’s k-closure for G ⊆ Sn Aut Inv(k) G ( Wielandt’s notation: gp(k- rel G)) Then, for G ≤ Sn, Aut Inv(2) G ⊇ · · · ⊇ Aut Inv(k) G ⊇ · · · ⊇ Aut Inv(n) G = G

Example

The general linear group GL(n, q) is 3-closed in Wielandt’s sense, as it is definable by ternary and binary relations: ̺+ = {(x, y, z) | z = x + y} , ̺c = {(x, y) | y = c · x} (c ∈ GF(q)) .

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (32/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Connections between the closures

What is the connection between the two closures: G

(k) (invariance

groups of functions f : kn → k) and Aut Inv(k) G (automorphism groups of k-ary relations)?

Proposition

For all G ≤ Sn we have G

(k+1) ⊆ Aut Inv(k) G. In particular, every

k-closed group G is Galois closed over k + 1.

(Thus every group which is not Galois closed over k cannot be characterized by (k−1)-ary relations.)

Proposition

For all G ≤ Sn we have G

(2) = G =

⇒ Aut Inv(⌊ n

2 ⌋) G = G.

Example

The Mathieu group M12 ≤ S12 satisfies M12

(2) = M12, as it is the

automorphism group of a hypergraph. Therefore Aut Inv(6) M12 = M12. On the other hand, M12 is 5-transitive, hence Aut Inv(5) M12 = S12.

• R. P¨
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Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Problem

Find the smallest number w(n, k) such that for every G ≤ Sn we have G

(k) = G =

⇒ Aut Inv(w(k,n)) G = G. From the Proposition and the Mathieu group example we conclude w(2, n) ≤ n 2

• and w(2, 12) = 6.
• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (34/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

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´ ath, G. Makay, R. P¨

• schel, and T. Waldhauser, Invariance groups of finite

functions and orbit equivalence of permutation groups. Open Math. 13(1), (2015), 83–95. N.F.J. Inglis, On orbit equivalent permutation groups, Arch. Math. 43 (1984), 297–300.

• A. Kisielewicz, Symmetry groups of Boolean functions and constructions of permutation

groups, J. Algebra 199 (1998), 379–403. ´

• A. Seress, Primitive groups with no regular orbits on the set of subsets, Bull. Lond. Math.

Soc., 29 (1997), 697–704.

• J. Siemons, A. Wagner, On finite permutation groups with the same orbits on unordered

sets, Arch. Math. 45 (1985), 492–500.

• H. Wielandt, Permutation groups through invariant relations and invariant functions, Dept.
• f Mathematics, Ohio State University, 1969.
• W. Xiao,Linear symmetries of Boolean functions. Discrete Applied Mathematics 149,

(2005), 192–199.

• R. P¨
• schel, Invariance groups of functions, orbit equivalence and group actions (35/36)

Invariance groups General point of view Characterizations of the closure Concrete results permutations and relations

Additional references

• D. Betten, Geometrische Permutationsgruppen. Mitt. Math. Gesellsch. Hamburg 10(5),

(1977), 317–324. L.A. Kaluˇ znin and M.H. Klin, On some numerical invariants of permutation groups. Latvi˘ ısk. Mat. Eˇ zegodnik (Vyp. 18), (1976), 81–99, 222.

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