Regular dessins with nilpotent automorphism groups Hu Kan Zhejiang - - PowerPoint PPT Presentation

Regular dessins with nilpotent automorphism groups

Hu Kan

Zhejiang Ocean University

Joint work with Roman Nedela and Na-Er Wang Shanghai, China, 2015

What is a dessin?

Map an embedding i : Γ ֒ → C of a connected graph Γ, possibly with multiple edges and loops, into an

• rientable closed surface C, such that each

component of C\i(Γ) is homeomorphic to an open disc.

What is a dessin?

Map an embedding i : Γ ֒ → C of a connected graph Γ, possibly with multiple edges and loops, into an

• rientable closed surface C, such that each

component of C\i(Γ) is homeomorphic to an open disc. Dessin a 2-colored bipartite map, i.e., its underlying graph is a bipartite graph whose vertices are colored by a fixed coloring, say black and white, such that vertices

• f the same color are not adjacent.

Example (A map on the sphere)

Example (K3,3 embedded into torus)

The opposite sides of the outer rhombus are identified to form the torus.

Combinatorial description

Let D = (Γ, C) be a dessin, and let E denote the set of edges

• f D. Then

◮ by following the global orientation of C we obtain at each

vertex v a cycle which permutes locally the edges incident with v,

Combinatorial description

Let D = (Γ, C) be a dessin, and let E denote the set of edges

• f D. Then

◮ by following the global orientation of C we obtain at each

vertex v a cycle which permutes locally the edges incident with v,

◮ let ρ (resp. λ) be the product of the cycles at black vertices

(resp. at white vertices), then ρ and λ are permutations of E.

Combinatorial description

Let D = (Γ, C) be a dessin, and let E denote the set of edges

• f D. Then

◮ by following the global orientation of C we obtain at each

vertex v a cycle which permutes locally the edges incident with v,

◮ let ρ (resp. λ) be the product of the cycles at black vertices

(resp. at white vertices), then ρ and λ are permutations of E.

◮ let

Mon(D) = ρ, λ then the connectivity of Γ implies that Mon(D) ≤ Sym(E) is transitive on E.

Combinatorial description

Vice versa, given a two-generated transitive permutation group A = ρ, λ on a nonempty set E, one can reconstruct a dessin D as follows:

Combinatorial description

Vice versa, given a two-generated transitive permutation group A = ρ, λ on a nonempty set E, one can reconstruct a dessin D as follows:

◮ the elements of E are identified with the edges of D, and the

cycles of ρ and λ are identified with the black vertices and white vertices, with incidence given by containment. In this way we obtain a bipartite graph Γ.

Combinatorial description

Vice versa, given a two-generated transitive permutation group A = ρ, λ on a nonempty set E, one can reconstruct a dessin D as follows:

◮ the elements of E are identified with the edges of D, and the

cycles of ρ and λ are identified with the black vertices and white vertices, with incidence given by containment. In this way we obtain a bipartite graph Γ.

◮ the successive powers of ρ and λ give the rotation of edges

around each vertex, and these local orientations determine an embedding of Γ into an oriented surface.

Automorphism group

An automorphism of D is a permutation σ of E such that σρ = ρσ, σλ = λσ, and they form the automorphism group Aut(D) of D.

Automorphism group

An automorphism of D is a permutation σ of E such that σρ = ρσ, σλ = λσ, and they form the automorphism group Aut(D) of D. In other words, Aut(D) = CSym(E)(Mon(D)).

Automorphism group

An automorphism of D is a permutation σ of E such that σρ = ρσ, σλ = λσ, and they form the automorphism group Aut(D) of D. In other words, Aut(D) = CSym(E)(Mon(D)).

Theorem

The automorphism group G = Aut(D) of a dessin acts semi-regularly on E, i.e., Ge = id for each e ∈ E.

Regular dessin the action of Aut(D) on E is transitive, and hence regular.

Regular dessin the action of Aut(D) on E is transitive, and hence regular. Type of regular dessin in a regular dessin D, all black vertices have the same valency l, all white vertices have the same valency m and all faces are bounded by 2n-gons, and the triple (l, m, n) is type of D.

Regular dessin the action of Aut(D) on E is transitive, and hence regular. Type of regular dessin in a regular dessin D, all black vertices have the same valency l, all white vertices have the same valency m and all faces are bounded by 2n-gons, and the triple (l, m, n) is type of D. Genus of regular dessin a regular dessin of type (l, m, n) has genus given by the Euler-Poincar´ e formula: 2 − 2g = χ = |E|(1 l + 1 m + 1 n − 1).

Theorem

The following are equivalent:

◮ D is a regular dessin.

Theorem

The following are equivalent:

◮ D is a regular dessin. ◮ Mon(D) is regular on E.

Theorem

The following are equivalent:

◮ D is a regular dessin. ◮ Mon(D) is regular on E. ◮ |Aut(D)| = |E|.

Theorem

The following are equivalent:

◮ D is a regular dessin. ◮ Mon(D) is regular on E. ◮ |Aut(D)| = |E|. ◮ Mon(D) ∼

= Aut(D).

Example (K3,3 revisited)

◮ K3,3 is a regular dessin of type (3, 3, 3) and genus 1.

Example (K3,3 revisited)

◮ K3,3 is a regular dessin of type (3, 3, 3) and genus 1. ◮ Aut(K3,3) = x, y | x3 = y3 = [x, y] = 1 ∼

= C3 × C3.

More examples of regular dessins

Example (Cube on the sphere)

◮ A regular dessin of type (3, 3, 2) and genus 0; ◮ Aut(D) = x, y | x3 = y3 = (xy)2 = 1 ∼

= A4.

Example (Cube on the torus)

Opposite sides of the outer hexagon are identified to form a torus

◮ A regular dessin of type (3, 3, 3) and genus 1. ◮ Aut(D) = x, y | x3 = y3 = (xy)3 = (x−1y)2 = 1 ∼

= A4.

Belyˇ ı’s Theorem

Theorem (Belyˇ ı, 1979)

A Riemann surface C, regarded as an algebraic curve, can be defined over the field ¯ Q of algebraic numbers iff there exists a non-constant meromorphic function β : C → ¯ C unramified outside {0, 1, ∞} where ¯ C = C ∪ {∞} is the Riemann sphere.

Dessins arising from Belyˇ ı functions

The trivial dessin B2 lifts along the Belyˇ ı function β to a dessin D on C, where

◮ the embedded graph is the preimage β−1[0, 1] of the closed

interval [0, 1] with black vertices β−1(0) and white vertices β−1(1),

Dessins arising from Belyˇ ı functions

The trivial dessin B2 lifts along the Belyˇ ı function β to a dessin D on C, where

◮ the embedded graph is the preimage β−1[0, 1] of the closed

interval [0, 1] with black vertices β−1(0) and white vertices β−1(1),

◮ the faces are the components of C\β−1[0, 1].

Example

Let C be the Fermat curve C = F3,3 given by x3 + y3 = 1. The function β : (x, y) → x3. is a Belyˇ ı function of degree 9 branched over 0, 1 and ∞, corresponding to the dessin K3,3 on C given above.

Example (K3,3 on torus as a 9-sheeted covering of B2)

β K3,3 B2

Galois operations

The absolute Galois group G = Gal(¯ Q/Q) acts naturally on the coefficients of polynomials and rational functions defining the algebraic curves and Belyˇ ı functions. This induces an action of G on dessins.

Galois operations

The absolute Galois group G = Gal(¯ Q/Q) acts naturally on the coefficients of polynomials and rational functions defining the algebraic curves and Belyˇ ı functions. This induces an action of G on dessins. In 1984 Grothendieck proposed to study G through its action

• n dessins.

Invariants of G

Theorem (Jones, Streit, 1997)

The following properties of a dessin are invariant under G:

◮ number of edges;

Invariants of G

Theorem (Jones, Streit, 1997)

The following properties of a dessin are invariant under G:

◮ number of edges; ◮ valency distributions of white and black vertices and faces;

Invariants of G

Theorem (Jones, Streit, 1997)

The following properties of a dessin are invariant under G:

◮ number of edges; ◮ valency distributions of white and black vertices and faces; ◮ type and genus;

Invariants of G

Theorem (Jones, Streit, 1997)

The following properties of a dessin are invariant under G:

◮ number of edges; ◮ valency distributions of white and black vertices and faces; ◮ type and genus; ◮ monodromy group and automorphism group.

Faithful actions of G

Theorem

G acts faithfully on (isomorphism classes of)

◮ dessins ( Grothendieck, 1984);

Faithful actions of G

Theorem

G acts faithfully on (isomorphism classes of)

◮ dessins ( Grothendieck, 1984); ◮ plane trees, i.e., one-face maps on the sphere (Schneps, 1994);

Faithful actions of G

Theorem

G acts faithfully on (isomorphism classes of)

◮ dessins ( Grothendieck, 1984); ◮ plane trees, i.e., one-face maps on the sphere (Schneps, 1994); ◮ dessins of a given genus (Girondo and Gonz´

alez-Diez, 2007);

Faithful actions of G

Theorem

G acts faithfully on (isomorphism classes of)

◮ dessins ( Grothendieck, 1984); ◮ plane trees, i.e., one-face maps on the sphere (Schneps, 1994); ◮ dessins of a given genus (Girondo and Gonz´

alez-Diez, 2007);

◮ regular dessins (Gonz´

alez-Diez, Jaikin-Zapirain, 2012).

Problem

Classify regular dessins.

Problem

Classify regular dessins.

◮ Classify regular dessins on a surface of a given genus;

Problem

Classify regular dessins.

◮ Classify regular dessins on a surface of a given genus; ◮ Classify regular dessins with a given underlying graph;

Problem

Classify regular dessins.

◮ Classify regular dessins on a surface of a given genus; ◮ Classify regular dessins with a given underlying graph; ◮ Classify regular dessins with a given automorphism group.

Algebraic description

Each dessin D can be regarded as a transitive permutation representation θ : F2 → A = Mon(D), X → ρ, Y → λ, where F2 = X, Y | − is the free group of rank two.

Algebraic description

Each dessin D can be regarded as a transitive permutation representation θ : F2 → A = Mon(D), X → ρ, Y → λ, where F2 = X, Y | − is the free group of rank two. Define N = θ−1(Ae), e ∈ E. Then the subgroup N is unique up to conjugacy, and is called the dessin subgroup of D.

Algebraic description

Each dessin D can be regarded as a transitive permutation representation θ : F2 → A = Mon(D), X → ρ, Y → λ, where F2 = X, Y | − is the free group of rank two. Define N = θ−1(Ae), e ∈ E. Then the subgroup N is unique up to conjugacy, and is called the dessin subgroup of D.

Theorem

A dessin D is regular iff the associated dessin subgroup N is normal in F2, in which case Aut(D) ∼ = F2/N.

Theorem

The set R(G) of isomorphism classes of regular dessins with a given automorphism group G corresponds bijectively

◮ to the set N(G) of normal subgroups N of F2 such that

G = F2/N,

Theorem

The set R(G) of isomorphism classes of regular dessins with a given automorphism group G corresponds bijectively

◮ to the set N(G) of normal subgroups N of F2 such that

G = F2/N,

◮ or to the orbits of Aut(G) on the set Ω(G) of generating

pairs (x, y) of G, i.e., Ω(G) = {(x, y) | G = x, y}.

Theorem

The set R(G) of isomorphism classes of regular dessins with a given automorphism group G corresponds bijectively

◮ to the set N(G) of normal subgroups N of F2 such that

G = F2/N,

◮ or to the orbits of Aut(G) on the set Ω(G) of generating

pairs (x, y) of G, i.e., Ω(G) = {(x, y) | G = x, y}.

Corollary

Let G be a finite group. Then the number r(G) of isomorphism classes of regular dessin D with Aut(D) ∼ = G is r(G) = |Ω(G)| |Aut(G)|.

Example

◮ Let G = Cn be the cyclic group of order n, then r(G) = ψ(n)

where ψ(n) = n

• p|n

(1 + 1 p) is the Dedekind’s totient function.

Example

◮ Let G = Cn be the cyclic group of order n, then r(G) = ψ(n)

where ψ(n) = n

• p|n

(1 + 1 p) is the Dedekind’s totient function.

◮ Let G = Cn × Cm where m|n, then r(G) = ψ(n/m).

Example

◮ Let G = Cn be the cyclic group of order n, then r(G) = ψ(n)

where ψ(n) = n

• p|n

(1 + 1 p) is the Dedekind’s totient function.

◮ Let G = Cn × Cm where m|n, then r(G) = ψ(n/m). ◮ Let G = D2n, n ≥ 3, then r(G) = 3.

Example

◮ Let G = Cn be the cyclic group of order n, then r(G) = ψ(n)

where ψ(n) = n

• p|n

(1 + 1 p) is the Dedekind’s totient function.

◮ Let G = Cn × Cm where m|n, then r(G) = ψ(n/m). ◮ Let G = D2n, n ≥ 3, then r(G) = 3. ◮ Let G = A5, then r(G) = 19.

Example

◮ Let G = Cn be the cyclic group of order n, then r(G) = ψ(n)

where ψ(n) = n

• p|n

(1 + 1 p) is the Dedekind’s totient function.

◮ Let G = Cn × Cm where m|n, then r(G) = ψ(n/m). ◮ Let G = D2n, n ≥ 3, then r(G) = 3. ◮ Let G = A5, then r(G) = 19. ◮ Let G = Q8, then r(G) = 1.

Dessin operations

Let D be a dessin, and N ≤ F2 be the associated dessin

• subgroup. Then

◮ each τ ∈ Aut(F2) transforms N to Nτ, and hence transforms

D to a dessin Dτ.

Dessin operations

Let D be a dessin, and N ≤ F2 be the associated dessin

• subgroup. Then

◮ each τ ∈ Aut(F2) transforms N to Nτ, and hence transforms

D to a dessin Dτ.

◮ if τ ∈ Inn(F2), then N is conjugate to Nτ, and hence D ∼

= Dτ.

Dessin operations

Let D be a dessin, and N ≤ F2 be the associated dessin

• subgroup. Then

◮ each τ ∈ Aut(F2) transforms N to Nτ, and hence transforms

D to a dessin Dτ.

◮ if τ ∈ Inn(F2), then N is conjugate to Nτ, and hence D ∼

= Dτ.

◮ so the outer automorphism

Out(F2) = Aut(F2) Inn(F2) acts on the isomorphism classes of dessins, and it is called the group of dessin operations.

Invariants of Out(F2)

Up to group isomorphism,

◮ the monodromy group of a dessin D is invariant under

Out(F2);

◮ the automorphism group of D is invariant under Out(F2).

External symmetry

External symmetry D possesses an external symmetry σ if D ∼ = Dσ where σ ∈ Aut(F2)\Inn(F2).

External symmetry

External symmetry D possesses an external symmetry σ if D ∼ = Dσ where σ ∈ Aut(F2)\Inn(F2). Symmetric dessin D ∼ = Dσ where σ : X → Y , Y → X.

External symmetry

External symmetry D possesses an external symmetry σ if D ∼ = Dσ where σ ∈ Aut(F2)\Inn(F2). Symmetric dessin D ∼ = Dσ where σ : X → Y , Y → X. Reflexible dessin D ∼ = Dσ where σ : X → X −1, Y → Y −1.

External symmetry

External symmetry D possesses an external symmetry σ if D ∼ = Dσ where σ ∈ Aut(F2)\Inn(F2). Symmetric dessin D ∼ = Dσ where σ : X → Y , Y → X. Reflexible dessin D ∼ = Dσ where σ : X → X −1, Y → Y −1. Self-Petrie-dual dessin D ∼ = Dτ where σ : X → X −1, Y → Y .

External symmetry

External symmetry D possesses an external symmetry σ if D ∼ = Dσ where σ ∈ Aut(F2)\Inn(F2). Symmetric dessin D ∼ = Dσ where σ : X → Y , Y → X. Reflexible dessin D ∼ = Dσ where σ : X → X −1, Y → Y −1. Self-Petrie-dual dessin D ∼ = Dτ where σ : X → X −1, Y → Y . Totally symmetric dessin a regular dessin which is invariant under all dessin operations.

Example

◮ The cube on the sphere is symmetric and reflexible, but not

self-Petrie-dual.

Example

◮ The cube on the sphere is symmetric and reflexible, but not

self-Petrie-dual.

◮ The cube on the torus is symmetric and reflexible, but not

self-Petrie-dual.

Example

◮ The cube on the sphere is symmetric and reflexible, but not

self-Petrie-dual.

◮ The cube on the torus is symmetric and reflexible, but not

self-Petrie-dual. In fact, each of them has the other as its Petrie-dual.

Petrie-dual of cube

Example (Cube on the sphere revisited)

A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.

Petrie-dual of cube

Example (Cube on the sphere revisited)

A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.

Petrie-dual of cube

Example (Cube on the sphere revisited)

A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.

Petrie-dual of cube

Example (Cube on the sphere revisited)

A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.

Petrie-dual of cube

Example (Cube on the sphere revisited)

A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.

Petrie-dual of cube

Example (Cube on the sphere revisited)

A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.

Petrie-dual of cube

Example (Cube on the sphere revisited)

A zig-zag path in the cube on the sphere. Cut the cube along all zig-zag pathes and glue them together along the new face boundaries we get the cube embedded into the torus.

The universal dessin

Let G be a finite 2-generator group, define K(G) =

• N∈N(G)

N.

The universal dessin

Let G be a finite 2-generator group, define K(G) =

• N∈N(G)

N. Then K(G) is the intersection of finitely many normal subgroups

• f finite index in F2, and hence it is normal of finite index in F2 as

well.

The universal dessin

Let G be a finite 2-generator group, define K(G) =

• N∈N(G)

N. Then K(G) is the intersection of finitely many normal subgroups

• f finite index in F2, and hence it is normal of finite index in F2 as

well. Define U(G) to be the regular dessin correponding to K(G), and ˆ G = F2/K(G).

Theorem (Jones, 2013)

◮ U(G) is the smallest regular dessin which covers all regular

dessins in R(G).

Theorem (Jones, 2013)

◮ U(G) is the smallest regular dessin which covers all regular

dessins in R(G).

◮ The group ˆ

G underlies a unique regular dessin, i.e., the dessin U(G).

Theorem (Jones, 2013)

◮ U(G) is the smallest regular dessin which covers all regular

dessins in R(G).

◮ The group ˆ

G underlies a unique regular dessin, i.e., the dessin U(G).

◮ The dessin U(G) is totally symmetric.

Theorem (Jones, 2013)

◮ U(G) is the smallest regular dessin which covers all regular

dessins in R(G).

◮ The group ˆ

G underlies a unique regular dessin, i.e., the dessin U(G).

◮ The dessin U(G) is totally symmetric. ◮ The dessin U(G) is defined over the field Q of rational

numbers.

Theorem (Jones, 2013)

◮ U(G) is the smallest regular dessin which covers all regular

dessins in R(G).

◮ The group ˆ

G underlies a unique regular dessin, i.e., the dessin U(G).

◮ The dessin U(G) is totally symmetric. ◮ The dessin U(G) is defined over the field Q of rational

numbers.

◮ If G is non-abelian simple, then ˆ

G = G r, where r = r(G) and G r is the rth direct product of G.

Theorem (Jones, 2013)

◮ U(G) is the smallest regular dessin which covers all regular

dessins in R(G).

◮ The group ˆ

G underlies a unique regular dessin, i.e., the dessin U(G).

◮ The dessin U(G) is totally symmetric. ◮ The dessin U(G) is defined over the field Q of rational

numbers.

◮ If G is non-abelian simple, then ˆ

G = G r, where r = r(G) and G r is the rth direct product of G.

◮ If G is solvable of derived length d, then so is ˆ

G.

Theorem (Jones, 2013)

◮ U(G) is the smallest regular dessin which covers all regular

dessins in R(G).

◮ The group ˆ

G underlies a unique regular dessin, i.e., the dessin U(G).

◮ The dessin U(G) is totally symmetric. ◮ The dessin U(G) is defined over the field Q of rational

numbers.

◮ If G is non-abelian simple, then ˆ

G = G r, where r = r(G) and G r is the rth direct product of G.

◮ If G is solvable of derived length d, then so is ˆ

G.

◮ If G is nilpotent of nilpotence class c, then so is ˆ

G.

Example (Jones, 2013)

◮ If G = Cn, then ˆ

G = Cn × Cn. In fact, if G = Cn × Cm, m|n, then ˆ G = Cn × Cn.

Example (Jones, 2013)

◮ If G = Cn, then ˆ

G = Cn × Cn. In fact, if G = Cn × Cm, m|n, then ˆ G = Cn × Cn.

◮ If G = A5, then ˆ

G = G 19.

Problem

Classify finite groups which underlie a unique regular dessin.

Problem

Classify finite groups which underlie a unique regular dessin. A a subproblem, we are interested in

Problem

Classify finite nilpotent groups which underlie a unique regular dessin.

Problem

Classify finite groups which underlie a unique regular dessin. A a subproblem, we are interested in

Problem

Classify finite nilpotent groups which underlie a unique regular dessin. Since every finite nilpotent group is the direct product of its Sylow subgroups, the problem is reduced to

Problem

Classify finite p-groups which underly a unique regular dessin.

Example

Let p be an odd prime and G be the non-abelian non-metacyclic p-group of order p3, that is, G = x, y|xp = yp = zp = [x, z] = [y, z] = 1, z := [x, y] .

Example

Let p be an odd prime and G be the non-abelian non-metacyclic p-group of order p3, that is, G = x, y|xp = yp = zp = [x, z] = [y, z] = 1, z := [x, y] . Then G underlies a unique regular dessin.

Example

Let p be an odd prime and G be the non-abelian non-metacyclic p-group of order p3, that is, G = x, y|xp = yp = zp = [x, z] = [y, z] = 1, z := [x, y] . Then G underlies a unique regular dessin. For instance when p = 3, the dessin is a regular embedding of the Pappus graph into the torus.

Pappus graph on the torus

Example

Main results

Theorem (H., Roman Nedela, Na-Er Wang, 2014)

A finite p-group G of class at most three which underlies a unique regular dessin is isomorphic to one of the following groups: (A) A single family of class c(G) = 1: G = x, y | xpa = ypa = [x, y] = 1 ∼ = Cpa × Cpa, a ≥ 0.

Continued

(B) Three families of class c(G) = 2:

(1) p > 2 and 1 ≤ b ≤ a, G = x, y | xpa = y pa = zpb = [x, z] = [y, z] = 1, z := [x, y] . (2) p = 2 and 1 ≤ b ≤ a − 1, G = x, y | x2a = y 2a = z2b = [x, z] = [y, z] = 1, z := [x, y] . (3) p = 2 and a ≥ 2, G = x, y | x2a = [x, z] = [y, z] = 1, x2a−1 = y 2a−1 = z2a−2, z := [x, y]

Continued

(C) Six families of class c(G) = 3:

(1) p = 3 and 1 ≤ c < b = a or 1 ≤ c ≤ b ≤ a − 1, G = x, y|x3a = y 3a = z3b = u3c = v 3c = [x, u] = [x, v] = [y, u] = [y, v] = 1, z := [x, y], u := [z, x], v := [z, y] . (2) p > 3 and 1 ≤ c ≤ b ≤ a, G = x, y|xpa = y pa = zpb = upc = v pc = [x, u] = [x, v] = [y, u] = [y, v] = 1, z := [x, y], u := [z, x], v := [z, y] . (3) p = 2 and 1 ≤ c ≤ b ≤ a − 1, G = x, y|x2a = y 2a = z2b = u2c = v 2c = [x, u] = [x, v] = [y, u] = [y, v] = 1, z := [x, y], u := [z, x], v := [z, y] .

Continued

(4) p = 2 and 1 ≤ c ≤ b ≤ a − 2, G = x, y|x2a = y2a = z2b = u2c = v2c = [x, u] = [x, v] = [y, u] = [y, v] = 1, x2a−1 = u2c−1, y2a−1 = v2c−1, z := [x, y], u := [z, x], v := [z, y] . (5) p = 2 and 1 ≤ c ≤ b ≤ a − 1, G = x, y|x2a = y2a = z2b = u2c = v2c = [x, u] = [x, v] = [y, u] = [y, v] = 1, x2a−1 = z2b−1, y2a−1 = z2b−1, z := [x, y], u := [z, x], v := [z, y] .

Continued

(6) p = 2 and 1 ≤ c ≤ a − 2, G = x, y|x2a = y2a = z2a−1 = u2c = v2c = [x, u] = [x, v] = [y, u] = [y, v] = 1, x2a−1 = z2a−2u2c−1, y2a−1 = z2a−2v2c−1, z := [x, y], u := [z, x], v := [z, y] . Moreover, the groups from distinct families, or from the same family but with distinct parameters, are pairwise non-isomorphic.

Remark

In “Groups of Prime Power Order, 2008”, Y. Berkovich and Z. Janko posed a problem of studying p-group G such that |G : Φ(G)| = pd and |Aut(G)| = (pd − 1) . . . (pd − pd−1)|Φ(G)|d, that is, G is a d-generator p-group and its automorphism group Aut(G) acts transitively on its generating d-tuples; see Research Problems and Themes I 35(a). Our main result solves this problem when d = 2 and c(G) ≤ 3.

• G. Gonz´

alez-Diez, A. Jaikin-Zapirain, The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces, preprint, 2013.

• A. Grothendieck, Esquisse d’un programme, preprint, 1984.
• K. Hu, R. Nedela, N.-E. Wang, Nilpotent groups of class two

which underly a unique regular dessin, Geometriae Dedicate,

• DOI. 10.1007/s10711-015-0074-8
• K. Hu, R. Nedela, N.-E. Wang, Nilpotent groups of class three

which underly a unique regular dessin, submitted. G.A. Jones, Regular dessins with a given automorphism group, arXiv:1309.5219 [math.GR], 2013. G.A. Jones, D. Singerman, Belyˇ ı functions, hypermaps and Galois groups, Bull. London Math. Soc. 28 (1996) 561–590.

Thank you for your attention.