Abstract regular polytopes acting as transitive subgroups of Sym(n)

Mark Mixer

Wentworth Institute of Technology

February 14 2015

Mark Mixer

Abstract regular polytopes acting as transitive subgroups of Sym(n)

Mark Mixer

Wentworth Institute of Technology

February 14 2015

Mark Mixer

Abstract regular polytopes acting as transitive subgroups of Sym(n)

Mark Mixer

Wentworth Institute of Technology

Happy Valentine’s Day 2015

Mark Mixer

Platonic Solids

Mark Mixer

Outline

Introduction

Regular polytopes and string C-groups

String C-groups acting on sets

Permutation graphs

Fracture Graphs

Ideas and Results in High Ranks

Mark Mixer

Platonic Solids and Groups

[5, 3] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)5 = (ρ1ρ2)3〉

[3, 5] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)3 = (ρ1ρ2)5〉[3, 3] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)3 = (ρ1ρ2)3〉[3, 4] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)3 = (ρ1ρ2)4〉[4, 3] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)4 = (ρ1ρ2)3〉

Mark Mixer

Platonic Solids and Groups

[5, 3] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)5 = (ρ1ρ2)3〉[3, 5] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)3 = (ρ1ρ2)5〉[3, 3] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)3 = (ρ1ρ2)3〉[3, 4] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)3 = (ρ1ρ2)4〉[4, 3] = 〈ρ0, ρ1, ρ2 | ρ2i = 1 = (ρ0ρ2)2 = (ρ0ρ1)4 = (ρ1ρ2)3〉

Mark Mixer

String C-groups acting

Let Γ be a (string C) group, and let it act on the set[1, . . . , n].

[1, . . . , n] = flags of the polytope or elements of Γ

ManiplexesMonodromyCaley GraphsColorful Polytopes

This makes n really big.

[1, . . . , n] = orbits of flags of the polytope.

Symmetry type graphs

This makes n small, but we lose the group structure in thegraph.

Mark Mixer

String C-groups acting

Let Γ be a (string C) group, and let it act on the set[1, . . . , n].

[1, . . . , n] = flags of the polytope or elements of Γ

ManiplexesMonodromyCaley GraphsColorful Polytopes

This makes n really big.

[1, . . . , n] = orbits of flags of the polytope.

Symmetry type graphs

This makes n small, but we lose the group structure in thegraph.

Mark Mixer

String C-groups acting

Let Γ be a (string C) group, and let it act on the set[1, . . . , n].

[1, . . . , n] = flags of the polytope or elements of Γ

ManiplexesMonodromyCaley GraphsColorful Polytopes

This makes n really big.

[1, . . . , n] = orbits of flags of the polytope.

Symmetry type graphs

This makes n small

, but we lose the group structure in thegraph.

Mark Mixer

String C-groups acting

Let Γ be a (string C) group, and let it act on the set[1, . . . , n].

[1, . . . , n] = flags of the polytope or elements of Γ

ManiplexesMonodromyCaley GraphsColorful Polytopes

This makes n really big.

[1, . . . , n] = orbits of flags of the polytope.

Symmetry type graphs

This makes n small, but we lose the group structure in thegraph.

Mark Mixer

String C-groups acting

Let n be as small as possible so that the action of Γ on [1, . . . , n] isfaithful.

A string C-group Γ will be of “high rank” if its rank is “close” to n.

Example 1: High rank

Let Γ be the automorphism group of the regular tetrahedron [3, 3].Then r = 3 and n = 4 where [1, . . . , n] can be the vertices of thetetrahedron.

Mark Mixer

String C-groups acting

Let n be as small as possible so that the action of Γ on [1, . . . , n] isfaithful.

A string C-group Γ will be of “high rank” if its rank is “close” to n.

Example 1: High rank

Let Γ be the automorphism group of the regular tetrahedron [3, 3].Then r = 3 and n = 4 where [1, . . . , n] can be the vertices of thetetrahedron.

Mark Mixer

String C-groups acting

Let n be as small as possible so that the action of Γ on [1, . . . , n] isfaithful.

A string C-group Γ will be of “high rank” if its rank is “close” to n.

Example 2: High rank

Let Γ be the automorphism group of the regular cube [4, 3].Then r = 3 and n = 6 where [1, . . . , n] can be the 2-faces of thecube.

Since Γ ∼= Sym(4)× 2 no smaller n will work.

Mark Mixer

String C-groups acting

Let n be as small as possible so that the action of Γ on [1, . . . , n] isfaithful.

A string C-group Γ will be of “high rank” if its rank is “close” to n.

Example 3: Not high rank

Let Γ be the O’Nan sporadic group.Then r = 4 and n = 122760.

Mark Mixer

Permutation Representation Graphs

Let Γ be a permutation group of degree n generated by involutionsρ0, . . . , ρr−1.

The graph X with vertices [1, . . . , n] and a, b ∈ E (X )⇔ aρi = bis called the permutation representation graph of Γ.

Note 1: This is not a new idea. Marston had already been studying“Schreier coset graphs and their applications” in 1992.

Note 2: These are edge labeled multigraphs. Where eachtransposition in the involution ρi gives one i-edge.

Mark Mixer

Symmetries of a cube acting on faces

Mark Mixer

Symmetries of a cube acting on faces

Mark Mixer

Symmetries of a cube acting on faces

Mark Mixer

Symmetries of a cube acting on faces

Mark Mixer

Symmetries of a cube acting on faces

Mark Mixer

Symmetries of a cube acting on faces

Mark Mixer

Symmetries of a cube acting on faces

Mark Mixer

Symmetries of a cube acting on faces

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Permutation Representation Graphs and Fracture Graphs

Let Γ = 〈ρ0, . . . , ρr−1〉 be a permutation group of degree ngenerated by involutions, and let X be its permutationrepresentation graph.

5, 6, 3

If Γi := 〈ρj | j 6= i〉 is intransitive for all i then we can define a“fracture graph” for Γ.

Mark Mixer

Fracture Graphs

Define F as follows from a permutation representation graph Xwith all Γi intransitive.

V (F ) = V (X )

|E (F )| = r where r is the rank of Γ

a, b ∈ E (F )⇒

aρi = baρ 6= b for all ρ ∈ Γi

Mark Mixer

Fracture Graphs

Define F as follows from a permutation representation graph Xwith all Γi intransitive.

V (F ) = V (X )

|E (F )| = r where r is the rank of Γ

a, b ∈ E (F )⇒aρi = baρ 6= b for all ρ ∈ Γi

Mark Mixer

Fracture Graphs

Mark Mixer

Fracture Graphs

Mark Mixer

Fracture Graphs

Mark Mixer

Fracture Graphs

Mark Mixer

Fracture Graphs

Mark Mixer

Fracture Graphs

Mark Mixer

Properties of Fracture Graphs

Lemma:

Let F be a fracture graph and X be a permutation representationgraph for a group of degree n generated by r involutions.

F contains no cycles.

F has n − r connected components.

If there is a multi-edge in X then the vertices are in differentconnected components of F

If there are two i-edges in X then all the vertices of theseedges are not in the same connected component of F .

Mark Mixer

High Rank Fracture Graphs

Lemma:

Let F be a fracture graph and X be a permutation representationgraph for a group of degree n generated by r involutions.

F contains no cycles.

F has n − r connected components.

If there is a multi-edge in X then the vertices are in differentconnected components of F

If there are two i-edges in X then all the vertices of theseedges are not in the same connected component of F .

Let r = n − 1.

F is a tree.

X has no multi-edges.

There are not two i edges in X for any i .

Mark Mixer

High Rank Fracture Graphs

Lemma:

Let F be a fracture graph and X be a permutation representationgraph for a group of degree n generated by r involutions.

F contains no cycles.

F has n − r connected components.

If there is a multi-edge in X then the vertices are in differentconnected components of F

If there are two i-edges in X then all the vertices of theseedges are not in the same connected component of F .

Let r = n − 1.

F is a tree.

X has no multi-edges.

There are not two i edges in X for any i .

Mark Mixer

High Rank Fracture Graphs

Lemma:

Let F be a fracture graph and X be a permutation representationgraph for a group of degree n generated by r involutions.

F contains no cycles.

F has n − r connected components.

If there is a multi-edge in X then the vertices are in differentconnected components of F

If there are two i-edges in X then all the vertices of theseedges are not in the same connected component of F .

Let r = n − 1.

F is a tree.

X has no multi-edges.

There are not two i edges in X for any i .

Mark Mixer

High Rank Fracture Graphs

Lemma:

Let F be a fracture graph and X be a permutation representationgraph for a group of degree n generated by r involutions.

F contains no cycles.

F has n − r connected components.

If there is a multi-edge in X then the vertices are in differentconnected components of F

If there are two i-edges in X then all the vertices of theseedges are not in the same connected component of F .

Let r = n − 1.

F is a tree.

X has no multi-edges.

There are not two i edges in X for any i .

Mark Mixer

High Rank Fracture Graphs

Lemma:

Let F be a fracture graph and X be a permutation representationgraph for a group of degree n generated by r involutions.

F contains no cycles.

F has n − r connected components.

If there is a multi-edge in X then the vertices are in differentconnected components of F

If there are two i-edges in X then all the vertices of theseedges are not in the same connected component of F .

Let r = n − 1.

F is a tree.

X has no multi-edges.

There are not two i edges in X for any i .

Mark Mixer

High Rank: r = n − 1

If r = n − 1 then X is a tree.

Additionally, if (ρiρj)2 for |i − j | > 1

Author's personal copy

M.E. Fernandes et al. / Journal of Combinatorial Theory, Series A 119 (2012) 42–56 45

Fig. 1. The CPR graph of the (n − 1)-simplex.

Proof of Lemma 2.2. Since Γ0 and Γr−1 are C-groups, by the above proposition, we just need to verifythat Γ0,r−1 = Γ0 ∩ Γr−1. It is always the case that Γ0,r−1 6 Γ0 ∩ Γr−1 6 Γ0, and since ρr−1 /∈ Γr−1, wehave Γ0,r−1 6 Γ0 ∩ Γr−1 Γ0. We get the desired equality since Γ0,r−1 is maximal in Γ0.

We use a notation similar to above for many groups. In general, for any group G and any generat-ing set g0, . . . , gk, let I = 0, . . . ,k, and define the group G j := ⟨gi | i ∈ I \ j⟩. Similarly, for lists ofsubscripts, we define G j1,..., jm := ⟨gi | i ∈ I \ j1, . . . , jm⟩.

3. Permutation representations

In this section we discuss permutation representations of string C-groups. In particular we de-fine CPR graphs for regular polytopes, introduced by Pellicer in [25] and [26], and we define asesqui-extension construction for extending a permutation representation of a string C-group whilemaintaining the string C-group structure. This construction can be seen as a special case of a Petrie-like construction described by Hartley and Leemans in [14], or as a mixing of two polytopes describedin [20] when the groups are string C-groups.

3.1. CPR graphs

We begin by recalling the basics of CPR graphs, referring to [26] for more details.A CPR graph of a regular d-polytope P is a permutation representation of Γ (P) = ⟨ρ0, . . . ,ρd−1⟩

represented on a graph as follows. Let φ be an embedding of Γ (P) into the symmetric group Sn forsome n. The CPR graph G of P determined by φ is the multigraph with n vertices, and with edgelabels in the set 0, . . . ,d − 1, such that any two vertices v, w are joined by an edge of label j if andonly if (v)(φ(ρ j)) = w . These representations are faithful since φ is an embedding.

For instance, take the symmetric group Sn with its natural action on a set Ω := 1, . . . ,n of npoints. Consider the string-C-group of the (n−1)-simplex Γ = ⟨ρ0, . . . ,ρn−2⟩ where ρi := (i +1, i +2)

with i ∈ 0, . . . ,n − 2. The CPR graph of Γ on n vertices is given in Fig. 1.The value of n for which we may choose an embedding into Sn is not unique. For example the

regular toroidal polytope P = 4,4(2,0) (see [23] for a description of this, and similar polytopes) has32 flags and thus there is an embedding of Γ (P) into S32, giving a CPR graph isomorphic to theCayley graph of Γ (P). However, the action of Γ (P) on the edges of the polytope determines theaction on the flags, and thus there is an embedding into S8 acting on the set of edges. From thetwo different embeddings, we get the two different CPR graphs in Fig. 2. Here ρ0, ρ1, and ρ2 arerepresented by the edges with no label, and with labels 1 and 2 respectively.

Since Γ (P) has a string diagram, the connected components of the graphs induced by edges withlabels i and j for |i − j| > 1 must either be single vertices, single edges, double edges, or alternatingsquares (see Fig. 3). This is due to the fact that if ρi and ρ j commute, we have ρ

ρ ji = ρi and therefore,

when conjugating ρi by ρ j , the set of edges corresponding to ρi in the CPR graph is stabilized. Inother words, ρ j must permute the edges corresponding to ρi .

We now recall a result that is useful in later sections. It allows us to avoid the use of a computerin proving that two groups are in fact C-groups.

Theorem 3.1. Let G be a connected, proper 3-edge labeled graph satisfying the conditions that each connectedcomponent induced by the edges with labels 0 and 2 is either a single vertex, a single edge, a double edge, oran alternating square. If G satisfies one of the three following conditions, then G is a CPR graph of a regular3-polytope.

then Γ ∼= Sym(n) and 〈ρ0, . . . , ρr−1〉 gives the Coxeter generatorsof the n − 1 simplex, if all Γi are intransitive.

Mark Mixer

High Rank: r = n − 1

If r = n − 1 then X is a tree.

Additionally, if (ρiρj)2 for |i − j | > 1

Author's personal copy

M.E. Fernandes et al. / Journal of Combinatorial Theory, Series A 119 (2012) 42–56 45

Fig. 1. The CPR graph of the (n − 1)-simplex.

Proof of Lemma 2.2. Since Γ0 and Γr−1 are C-groups, by the above proposition, we just need to verifythat Γ0,r−1 = Γ0 ∩ Γr−1. It is always the case that Γ0,r−1 6 Γ0 ∩ Γr−1 6 Γ0, and since ρr−1 /∈ Γr−1, wehave Γ0,r−1 6 Γ0 ∩ Γr−1 Γ0. We get the desired equality since Γ0,r−1 is maximal in Γ0.

We use a notation similar to above for many groups. In general, for any group G and any generat-ing set g0, . . . , gk, let I = 0, . . . ,k, and define the group G j := ⟨gi | i ∈ I \ j⟩. Similarly, for lists ofsubscripts, we define G j1,..., jm := ⟨gi | i ∈ I \ j1, . . . , jm⟩.

3. Permutation representations

In this section we discuss permutation representations of string C-groups. In particular we de-fine CPR graphs for regular polytopes, introduced by Pellicer in [25] and [26], and we define asesqui-extension construction for extending a permutation representation of a string C-group whilemaintaining the string C-group structure. This construction can be seen as a special case of a Petrie-like construction described by Hartley and Leemans in [14], or as a mixing of two polytopes describedin [20] when the groups are string C-groups.

3.1. CPR graphs

We begin by recalling the basics of CPR graphs, referring to [26] for more details.A CPR graph of a regular d-polytope P is a permutation representation of Γ (P) = ⟨ρ0, . . . ,ρd−1⟩

represented on a graph as follows. Let φ be an embedding of Γ (P) into the symmetric group Sn forsome n. The CPR graph G of P determined by φ is the multigraph with n vertices, and with edgelabels in the set 0, . . . ,d − 1, such that any two vertices v, w are joined by an edge of label j if andonly if (v)(φ(ρ j)) = w . These representations are faithful since φ is an embedding.

For instance, take the symmetric group Sn with its natural action on a set Ω := 1, . . . ,n of npoints. Consider the string-C-group of the (n−1)-simplex Γ = ⟨ρ0, . . . ,ρn−2⟩ where ρi := (i +1, i +2)

with i ∈ 0, . . . ,n − 2. The CPR graph of Γ on n vertices is given in Fig. 1.The value of n for which we may choose an embedding into Sn is not unique. For example the

regular toroidal polytope P = 4,4(2,0) (see [23] for a description of this, and similar polytopes) has32 flags and thus there is an embedding of Γ (P) into S32, giving a CPR graph isomorphic to theCayley graph of Γ (P). However, the action of Γ (P) on the edges of the polytope determines theaction on the flags, and thus there is an embedding into S8 acting on the set of edges. From thetwo different embeddings, we get the two different CPR graphs in Fig. 2. Here ρ0, ρ1, and ρ2 arerepresented by the edges with no label, and with labels 1 and 2 respectively.

Since Γ (P) has a string diagram, the connected components of the graphs induced by edges withlabels i and j for |i − j| > 1 must either be single vertices, single edges, double edges, or alternatingsquares (see Fig. 3). This is due to the fact that if ρi and ρ j commute, we have ρ

ρ ji = ρi and therefore,

when conjugating ρi by ρ j , the set of edges corresponding to ρi in the CPR graph is stabilized. Inother words, ρ j must permute the edges corresponding to ρi .

We now recall a result that is useful in later sections. It allows us to avoid the use of a computerin proving that two groups are in fact C-groups.

Theorem 3.1. Let G be a connected, proper 3-edge labeled graph satisfying the conditions that each connectedcomponent induced by the edges with labels 0 and 2 is either a single vertex, a single edge, a double edge, oran alternating square. If G satisfies one of the three following conditions, then G is a CPR graph of a regular3-polytope.

then Γ ∼= Sym(n) and 〈ρ0, . . . , ρr−1〉 gives the Coxeter generatorsof the n − 1 simplex

, if all Γi are intransitive.

Mark Mixer

High Rank: r = n − 1

If r = n − 1 then X is a tree.

Additionally, if (ρiρj)2 for |i − j | > 1

Author's personal copy

M.E. Fernandes et al. / Journal of Combinatorial Theory, Series A 119 (2012) 42–56 45

Fig. 1. The CPR graph of the (n − 1)-simplex.

Proof of Lemma 2.2. Since Γ0 and Γr−1 are C-groups, by the above proposition, we just need to verifythat Γ0,r−1 = Γ0 ∩ Γr−1. It is always the case that Γ0,r−1 6 Γ0 ∩ Γr−1 6 Γ0, and since ρr−1 /∈ Γr−1, wehave Γ0,r−1 6 Γ0 ∩ Γr−1 Γ0. We get the desired equality since Γ0,r−1 is maximal in Γ0.

We use a notation similar to above for many groups. In general, for any group G and any generat-ing set g0, . . . , gk, let I = 0, . . . ,k, and define the group G j := ⟨gi | i ∈ I \ j⟩. Similarly, for lists ofsubscripts, we define G j1,..., jm := ⟨gi | i ∈ I \ j1, . . . , jm⟩.

3. Permutation representations

In this section we discuss permutation representations of string C-groups. In particular we de-fine CPR graphs for regular polytopes, introduced by Pellicer in [25] and [26], and we define asesqui-extension construction for extending a permutation representation of a string C-group whilemaintaining the string C-group structure. This construction can be seen as a special case of a Petrie-like construction described by Hartley and Leemans in [14], or as a mixing of two polytopes describedin [20] when the groups are string C-groups.

3.1. CPR graphs

We begin by recalling the basics of CPR graphs, referring to [26] for more details.A CPR graph of a regular d-polytope P is a permutation representation of Γ (P) = ⟨ρ0, . . . ,ρd−1⟩

represented on a graph as follows. Let φ be an embedding of Γ (P) into the symmetric group Sn forsome n. The CPR graph G of P determined by φ is the multigraph with n vertices, and with edgelabels in the set 0, . . . ,d − 1, such that any two vertices v, w are joined by an edge of label j if andonly if (v)(φ(ρ j)) = w . These representations are faithful since φ is an embedding.

For instance, take the symmetric group Sn with its natural action on a set Ω := 1, . . . ,n of npoints. Consider the string-C-group of the (n−1)-simplex Γ = ⟨ρ0, . . . ,ρn−2⟩ where ρi := (i +1, i +2)

with i ∈ 0, . . . ,n − 2. The CPR graph of Γ on n vertices is given in Fig. 1.The value of n for which we may choose an embedding into Sn is not unique. For example the

regular toroidal polytope P = 4,4(2,0) (see [23] for a description of this, and similar polytopes) has32 flags and thus there is an embedding of Γ (P) into S32, giving a CPR graph isomorphic to theCayley graph of Γ (P). However, the action of Γ (P) on the edges of the polytope determines theaction on the flags, and thus there is an embedding into S8 acting on the set of edges. From thetwo different embeddings, we get the two different CPR graphs in Fig. 2. Here ρ0, ρ1, and ρ2 arerepresented by the edges with no label, and with labels 1 and 2 respectively.

Since Γ (P) has a string diagram, the connected components of the graphs induced by edges withlabels i and j for |i − j| > 1 must either be single vertices, single edges, double edges, or alternatingsquares (see Fig. 3). This is due to the fact that if ρi and ρ j commute, we have ρ

ρ ji = ρi and therefore,

when conjugating ρi by ρ j , the set of edges corresponding to ρi in the CPR graph is stabilized. Inother words, ρ j must permute the edges corresponding to ρi .

We now recall a result that is useful in later sections. It allows us to avoid the use of a computerin proving that two groups are in fact C-groups.

Theorem 3.1. Let G be a connected, proper 3-edge labeled graph satisfying the conditions that each connectedcomponent induced by the edges with labels 0 and 2 is either a single vertex, a single edge, a double edge, oran alternating square. If G satisfies one of the three following conditions, then G is a CPR graph of a regular3-polytope.

then Γ ∼= Sym(n) and 〈ρ0, . . . , ρr−1〉 gives the Coxeter generatorsof the n − 1 simplex, if all Γi are intransitive.

Mark Mixer

Abstract regular polytopes acting as transitive subgroupsof Sn

Theorem: Cameron, Fernandes, Leemans, M

Let Γ be a string C-group of rank r which is isomorphic to atransitive subgroup of Sn other than Sn or An. Then one of thefollowing holds:

1 r ≤ n/2;

2 n ≡ 2 mod 4, r = n/2 + 1 and Γ is 2 o Sn/2. The Schafli typeis [2, 3, . . . , 3, 4].

3 n = 6, 8, and Γ is one of four imprimitive examples.

4 n = 6, and Γ is PGL2(5) ∼= S5 (the 4-simplex).

Mark Mixer

Thanks and Happy Birthday!!!

Mark Mixer