The ranks of alternating string C-groups

TL;DR

This paper classifies all possible ranks of string C-group representations of alternating groups $A_n$ for $n geq 12$, providing a complete understanding of their structure within certain rank bounds.

Contribution

It determines the exact values of $n$ for which $A_n$ can be represented as a string C-group of a given rank between 3 and $n/2$.

Findings

Classifies string C-group ranks for all $A_n$, $n geq 12$

Provides explicit rank bounds for $A_n$ representations

Completes the classification for $A_n$ with $n eq 11$

Abstract

In this paper, string C-groups of all ranks $3 \leq r \leq \frac{n}{2}$ are provided for each alternating group $A_n$, $n \geq 12$. As the string C-group representations of $A_n$ have also been classified for $n \leq 11$, and it is known that larger ranks are impossible, this paper provides the exact values of $n$ for which $A_n$ can be represented as a string C-group of a fixed rank.

Equations

$$({\rho }_i{\rho }_j{)}^{p_{ij}}=ϵ(0\le i,j\le r-1),$$

$$p_{ij}=2\text{ for }|i-j|\ge 2.$$

$$⟨{\rho }_i\mid i\in I⟩\cap ⟨{\rho }_i\mid i\in J⟩=⟨{\rho }_i\mid i\in I\cap J⟩\text{ for }I,J\subseteq \{0,\dots ,r-1\}$$

$$({\rho }_2{\rho }_1{\rho }_0{\rho }_1{)}^2$$

Full text

1 Introduction

In Problem 32 of [18], Hartley asks “Find regular, chiral, or other polytopes whose automorphism groups are alternating groups $A_{n}$. In particular, given a rank $r$, for which $n$ does $A_{n}$ occur as the automorphism group of a regular or chiral polytope of rank $r$?” In [3], the maximum achievable rank for each group $A_{n}$ was found in the regular case. In this paper we finish the solution to Hartley’s question in the regular case, by finding string C-groups (regular polytopes) of all achievable ranks $r$ for each $n$.

The paper is organized as follows. In section 2 we briefly outline any necessary definitions and background. Sections 3 contains many families of string C-groups that are needed to prove the main theorem. Sections 4 and 5 consider string C-groups of rank at least seven, for odd and even $n$ respectively. In Section 6, we provide the string C-groups with ranks less than or equal to six for all possible $n$. Finally, in section 7, we summarize the main theorem.

2 Background and Basic Notions

The automorphism group of an abstract regular polytope, along with a distinguished set of generators $\{\rho_{0},\ldots,\rho_{r-1}\}$, is called a rank $r$ string C-group. In general, we say that a group $\Gamma$ is a rank $r$ string group generated by involutions (or an sggi for short) if $\Gamma$ is generated by $\{\rho_{0},\ldots,\rho_{r-1}\}$ which satisfy the following conditions.

[TABLE]

where $p_{ii}=1$ for all $i$, $2\leq p_{ji}=p_{ij}$ if $j=i-1$, and

[TABLE]

Moreover, if $\Gamma$ has the following intersection property, then it is considered to be a string C-group.

[TABLE]

The automorphism group $\Gamma(\mathcal{P})$ of an abstract regular polytope $\mathcal{P}$ is a string C-group, and conversely, it is known (see [12, Sec. 2E]) that an abstract regular $n$-polytope can be constructed uniquely from any string C-group.

We will often use the fact that not all of the intersections from Equation 3 need to be verified.

Proposition 2.1**.**

Let $\Gamma$ be a rank $r$ string group generated by involutions, and suppose that $\Gamma_{0}$ and $\Gamma_{r-1}$ are both string C-groups. If $\Gamma_{0}\cap\Gamma_{r-1}=\Gamma_{0,r-1}$, then $\Gamma$ is a string C-group. Moreover, if $\Gamma_{0,r-1}$ is a maximal subgroup of either $\Gamma_{0}$ or $\Gamma_{r-1}$ then this condition is satisfied.

Proof.

This combines Proposition 2E16 of [12] and Lemma 2.2 of [7]. ∎

Let $\Gamma=\langle\rho_{0},\,\ldots,\,\rho_{r-1}\rangle$ be a string group generated by involutions acting as a permutation group on a set $\{1,\,\ldots,\,n\}$. We can construct the permutation representation graph $\mathcal{X}$ of $\Gamma$ as the $r$-edge-labeled graph with $n$ vertices, and with a single $i$-edge $\{a,\,b\}$ whenever $a\rho_{i}=b$ with $a<b$. When $\Gamma$ is a string C-group that acts faithfully on $\{1,\ldots,n\}$, the graph $\mathcal{X}$ is called a CPR graph, as defined in [15].

If $P$ and $Q$ are string C-groups, then we say that $P$ covers $Q$ if there is a well-defined surjective homomorphism from $P$ to $Q$ that respects the canonical generators. In other words, if $P=\langle\rho_{0},\ldots,\rho_{r-1}\rangle$ and $Q=\langle\rho_{0}^{\prime},\ldots,\rho_{r-1}^{\prime}\rangle$, then $P$ covers $Q$ if there is a homomorphism that sends each $\rho_{i}$ to $\rho_{i}^{\prime}$.

Given string C-groups $P$ and $Q$, the mix of $P$ and $Q$, denoted $P\diamond Q$, is the subgroup of the direct product $P\times Q$ that is generated by the elements $(\rho_{i},\rho_{i}^{\prime})$. This group is the minimal string group generated by involutions that covers both $P$ and $Q$ - where again, we only consider homomorphisms that respect the generators; see [13, Section 5] for more details.

It is possible to mix a rank $r$ string C-group $P$ with a rank $s$ string C-group $Q$. In particular we often mix a string C-group with the automorphism group of an edge $e$ (which is a rank 1 regular polytope). To do so, we take $e=\langle\rho_{0},\ldots,\rho_{r-1}\rangle$ with defining relations $\rho_{0}^{2}=\epsilon$ and $\rho_{i}=\epsilon$ for $1\leq i\leq r-1$, and then use the same definition as before. In general, to mix two string C-groups of different ranks, we add trivial generators to the group of smaller rank.

The comix of $P$ and $Q$, denoted $P\boxempty Q$, is the largest string group generated by involutions that is covered by both $P$ and $Q$ [4]. A presentation for $P\boxempty Q$ can be obtained from that of $P$ by adding all of the relations of $Q$, rewriting the relations to use the generators of $P$ instead. The size of the comix of $P$ and $Q$ is the index of the mix in the full direct product. Throughout the paper we will rely on the following results about mixing of string C-groups.

Proposition 2.2**.**

(Theorem 5.12 of [13]) Suppose $P$ and $Q$ are rank $r$ string C-groups, and that $P_{r-1}$ covers $Q_{r-1}$ then $P\diamond Q$ is a string C-group.

Proposition 2.3**.**

(Theorem 7A7 of [12]) If $P$ is a rank $r$ string C-group then $P\diamond P_{r-1}$ is a string C-group.

Definition 2.4**.**

Let $\Gamma=\langle\rho_{0},\ldots,\rho_{r-1}\rangle$ be an sggi, and let $\tau$ be an involution in a supergroup of $\Gamma$ such that $\tau\not\in\Gamma$ and $\tau$ commutes with all of $\Gamma$. For fixed $k$, we define the group $\Gamma^{*}=\langle\rho_{i}\tau^{\eta_{i}}\,|\,i\in\{0,\,\ldots,\,r-1\}\rangle$ where $\eta_{i}=1$ if $i=k$ and 0 otherwise, the sesqui-extension of $\Gamma$ with respect to $\rho_{k}$ and $\tau$.

A sesqui-extension of a group $\Gamma$, with respect to its first generator can be seen as a mix of $\Gamma$ with the automorphism group of an edge, and thus we have the following.

Proposition 2.5**.**

*(Proposition 5.3 of [5])

If $\Psi$ is a sesqui-extension of a string C-group $\Gamma$ with respect to $\rho_{0}$, then $\Psi$ is a string C-group.*

The following lemma shows that we can often use this result in a more general setting.

Lemma 2.6**.**

(Lemma 5.4 of [5]) If $\Gamma=\left<\rho_{i}\,|\,i=0,\,\ldots,\,r-1\right>$ and $\Psi=\langle\rho_{i}\tau^{\eta_{i}}\,|\,i\in\{0,\,\ldots,\,r-1\}\rangle$ is a sesqui-extension of $\Gamma$ with respect to $\rho_{k}$, then:

$\Psi\cong\Gamma$* or $\Psi\cong\Gamma\times\langle\tau\rangle\cong\Gamma\times 2$.* 2. 2.

If the identity element of $\Gamma$ can be written as a product of generators involving an odd number of $\rho_{k}$’s, then $\Psi\cong\Gamma\times\langle\tau\rangle$. 3. 3.

if $\Gamma$ is a finite permutation group, $\tau$ and $\rho_{k}$ are odd permutations, and all other $\rho_{i}$ are even permutations, then $\Psi\cong\Gamma$. 4. 4.

whenever $\tau\notin\Psi$, $\Gamma$ is a string C-group if and only if $\Psi$ is a string C-group.

Proposition 2.7**.**

Let $\Gamma=\left<\rho_{i}\,|\,i=0,\,\ldots,\,r-1\right>$ and $\Psi=\langle\rho_{i}\tau^{\eta_{i}}\,|\,i\in\{0,\,\ldots,\,r-1\}\rangle$ be a sesqui-extension of $\Gamma$ with respect to $\rho_{k}$. If either $\Psi_{0}\cong\Gamma_{0}$ or $\Psi_{r-1}\cong\Gamma_{r-1}$ as string C-groups, then $\Psi$ is a string C-group.

Proof.

This is a consequence of part (b) of Proposition 2E16 in [12]. Assume $\Psi_{r-1}\cong\Gamma_{r-1}$ as string C-groups. The intersection condition of part (b) holds as $\tau$ is not in any of the groups $\Psi_{r-1}\cap\langle\rho_{k},\ldots,\rho_{r-1}\rangle$. ∎

For any permutation group $\Gamma$ of degree $n$, we will use the notation $S_{n}$ to represent the full symmetric group, $A_{n}$ to represent the full alternating group, and $G^{+}$ to denote $G\cap A_{n}$. If $\Gamma:=\langle\rho_{0},\ldots,\rho_{n-1}\rangle$, then for each $i$ we denote $\Gamma_{i}=\langle\rho_{j}\mid j\neq i\rangle$, where each $\Gamma_{i}$ is itself a string C-group. Similarly, we will denote $\Gamma_{i,j}=\langle\rho_{k}\mid k\not\in\{i,j\}\rangle$. The dual $\Gamma^{*}$ of a string C-group $\Gamma$ is the group generated by the same involutions, but with the indexing reversed.

Finally, we will occasionally use the following rank reduction technique of [2]. We will frequently need to apply this construction to the dual of a string C-group, and then take the dual again. When we do this, we simply call it the dual rank reduction.

Proposition 2.8**.**

Let $\Gamma=\langle\rho_{0},\ldots\rho_{r-1}\rangle$ be a rank $r$ string C-group, where $|\rho_{i}\rho_{i+1}|>2$ for all $0\leq i\leq r-2$. If $\rho_{0}\in\langle\rho_{0}\rho_{2},\rho_{3}\rangle$, then $\Gamma\cong\langle\rho_{1},\rho_{0}\rho_{2},\rho_{3},\ldots,\rho_{r-1}\rangle$ is a string C-group of rank $r-1$. Furthermore, if $\rho_{2}\rho_{3}$ has odd order, then this condition is satisfied.

Proof.

This combines the results of Theorem 1.1 and Corollary 1.2 of [2]. ∎

3 Building Blocks

In this section, we provide some examples of families of string C-groups which appear as subgroups of the groups in our main theorem.

Lemma 3.1**.**

For each $r\geq 3$ the permutation group $FL(r,k)$ given by the following graph is a string C-group isomorphic to $S_{r+1+k}$, for all odd $k\geq 0$, and

[TABLE]

for each $r\geq 3$ the permutation group $FL(r,k)$ given by the following graph is a string C-group isomorphic to $S_{r+1+k}$, for all even $k\geq 0$

[TABLE]

We point out that there are $k+2$ edges of labels 0 or 1 in each graph.

Proof.

This combines the results of Theorem 1, Theorem 2, and Lemma 21 from [6]. ∎

Lemma 3.2**.**

For each $r\geq 5$ the permutation group $R(r,k)$ given by the following graph is a string C-group isomorphic to $S_{r+3+k}$, for all odd $k\geq 0$, and

[TABLE]

for each $r\geq 5$ the permutation group $R(r,k)$ given by the following graph is a string C-group isomorphic to $S_{r+3+k}$, for all even $k\geq 0$.

[TABLE]

Proof.

We will prove this by induction on $k$. To clarify the notation, we point out that there are $k+2$ edges of labels 0 or 1 in each graph. When $k=0$, $R(r,k)$ is a string C-group isomorphic to $S_{r+3}$ as it is the dual of $FL(r,2)$ from Lemma 3.1. We will assume by induction that $R(r+1,k-1)$ is a string C-group isomorphic to $S_{r+1+3+k-1}$, and then applying the rank reduction from Proposition 2.8 to $R(r+1,k-1)$ we get $R(r,k)$, which is thus a string C-group isomorphic to $S_{r+3+k}$.

∎

Lemma 3.3**.**

For each $r\geq 4$ the permutation group $Sh(r,k)$ given by the following graph is a string C-group isomorphic to $(S_{2}\wr S_{r+\frac{k}{2}})^{+}$, for all $k\equiv 2\pmod{4}$, with $k\geq 0$.

[TABLE]

Note that there are $1+\frac{k}{2}$ edges of label 0 in each such graph. We also note that when $r=3$ this graph still provides a string C-group.

Proof.

We prove that the group is a string C-group by induction on $r$, where the proof of the base case will be nearly the same as the proof of the inductive step. Let $\Gamma=\langle\rho_{0},\ldots,\rho_{r-1}\rangle=Sh(r,k)$. When $r=4$, the group $\Gamma_{r-1}$ is a string C-group by Theorem 4.4 of [15]. Furthermore, when $r$ is greater than $4$, we can assume that $\Gamma_{r-1}$ is a string C-group as $\Gamma_{r-1}=Sh(r-1,k)$.

For all $r\geq 4$, the group $\Gamma_{0}$ is isomorphic to the string C-group $FL(r-1,\frac{k}{2})\diamond FL(r-1,\frac{k}{2})$ which is isomorphic to $S_{r+\frac{k}{2}}$. It remains to show that $\Gamma_{0}\cap\Gamma_{r-1}=\Gamma_{0,r-1}$.

The group $\Gamma$ is an imprimitive group with a natural block structure, having $r+\frac{k}{2}$ blocks of size two. The generator $\rho_{0}$ is the only generator acting within a block, and thus $\Gamma_{0}$ gives the action on the blocks.

Let $\alpha\in\Gamma_{0}\cap\Gamma_{r-1}$. Since $\alpha\in\Gamma_{0}$ it only acts on the blocks, and since $\alpha\in\Gamma_{r-1}$ it fixes the “last block” (the block in the support of $\rho_{r-1}$). Since $\Gamma_{0,r-1}$ gives the action on these $r+\frac{k}{2}-1$ blocks, we know that $\alpha\in\Gamma_{0,r-1}$. Thus by Proposition 2.1, $\Gamma$ is a string C-group. Finally, to prove that the group is the collection of all even permutations in the wreath product, observe that the group $\Gamma_{0}$ gives the full symmetric group acting on the blocks, and the element $(\rho_{0}\rho_{1})^{2}$ is a product of two disjoint transpositions, each swapping two elements within a block.

∎

Lemma 3.4**.**

For each $r\geq 6$ the permutation group $Bl(r,k)$ given by the following graph is a string C-group isomorphic to $(S_{2}\wr S_{r+2+\frac{k}{2}})^{+}$, for all $k\equiv 2\pmod{4}$, with $k\geq 0$.

[TABLE]

We note that there are $1+\frac{k}{2}$ edges of label 0 in each graph.

Proof.

The group $Bl(r,k)$ is obtained by applying the dual rank reduction of Lemma 2.8 to $Sh(r+2,k)$ and then again to the result. This shows that $Bl(r,k)$ is a string C-group, and also that $Bl(r,k)$ is isomorphic to $(S_{2}\wr S_{r+2+\frac{k}{2}})^{+}$. ∎

Lemma 3.5**.**

For each $r\geq 4$ the permutation group $P(r,k)$ given by the following graph is a string C-group isomorphic to $S_{r+2+k}$, for all odd $k\geq 0$, and

[TABLE]

for each $r\geq 4$ the permutation group $P(r,k)$ given by the following graph is a string C-group isomorphic to $S_{r+2+k}$, for all even $k\geq 0$, and

[TABLE]

There are $k+2$ edges of labels 0 or 1 in each graph. Also, note that when $r=3$ this graph still provides a string C-group, see for example Theorem 4.5 of [15] when $k$ is odd or Theorem 4.4 of [15] when $k$ is even.

Proof.

Let $\Gamma=\langle\rho_{0},\ldots,\rho_{r-1}\rangle=P(r,k)$. The group $\Gamma_{0}$ is a string C-group isomorphic to $S_{2}\times S_{r+1}$ as it is a sesqui-extension of string C-group $FL(r-1,1)$ from Lemma 3.1. The group $\Gamma_{r-1}$ is a string C-group isomorphic to $S_{2}\times S_{r+k}$ as it also is a sesqui-extension of string C-group $FL(r-1,k)$. Thus $\Gamma$ is isomorphic to $S_{r+2+k}$. Finally, the group $\Gamma_{0,r-1}$ is isomorphic to $S_{2}\times S_{r-1}\times S_{2}$, which is maximal in $\Gamma_{0}$, and thus, by Proposition 2.1, $\Gamma$ is also a string C-group. ∎

Lemma 3.6**.**

For each $r\geq 4$ the permutation group $Sp(r,k)$ given by the following graph is a string C-group isomorphic to $S_{r+2+\frac{k}{2}}\times S_{r+1+\frac{k}{2}}$ for $k\equiv 2\pmod{4}$ with $k\geq 0$.

[TABLE]

We note that there are $k+4$ edges of labels 0 or 1 in each such graph.

Proof.

Let $\Gamma=\langle\rho_{0},\ldots,\rho_{r-1}\rangle=Sp(r,k)$. Then, $\Gamma=P\diamond Q$, where $P=P(r,\frac{k}{2})$ from Lemma 3.5, and $Q$ is $FL(r,\frac{k}{2})$ from Lemma 3.1. Furthermore, the facets of $P$ cover the facets of $Q$; as the facets of $P$ are isomorphic to $S_{r+\frac{k}{2}}\times S_{2}$ where as the facets of $Q$ are isomorphic to $S_{r+\frac{k}{2}}$. Thus by Proposition 2.2, $\Gamma\cong P\diamond Q$ is a string C-group.

To determine that $\Gamma$ is the full direct product, we consider the comix of $P$ and $Q$, $C=P\boxempty Q$. We will write $C$ as generated by $\rho_{i}$, along with all the relations from both $P$ and $Q$. Thus, in $C$, $(\rho_{r-1}\rho_{r-2})^{3}=1$ from $Q$ and $1=(\rho_{r-1}\rho_{r-2})^{4}$ from $P$, and thus in $C$, $\rho_{r-1}=\rho_{r-2}$.

Then, it follows that in $C$, $(\rho_{r-2}\rho_{r-3})^{3}=1=(\rho_{r-1}\rho_{r-3})^{3}=(\rho_{r-1}\rho_{r-3})^{2}$, and thus $\rho_{r-1}=\rho_{r-3}$. Furthermore in $C$, we know that $(\rho_{r-3}\rho_{r-2}\rho_{r-1})^{5}=1$ from $P$, and so for instance $\rho_{r-1}^{5}=\rho_{r-1}^{2}=1$, and thus $\rho_{r-1}=\rho_{r-2}=\rho_{r-3}=1$.

Then, it will follow that all $\rho_{i}=1$ in $C$. For example in $C$, $(\rho_{r-3}\rho_{r-4})^{3}=1$ and thus in C $\rho_{r-4}=1$. This argument works for showing that in $C$, $1=\rho_{2}=\rho_{3}=\cdots=\rho_{r-1}$.

Finally, $(\rho_{0}\rho_{1}\rho_{2})^{\frac{k}{2}+4}=1$ and $(\rho_{0}\rho_{1})^{\frac{k}{2}+3}=1$, and so in $C$, $\rho_{0}=\rho_{1}$.

We have showed that $C=P\boxempty Q$ has size at most two, and therefore $P\diamond Q$ is the full direct product or an index two subgroup of the full direct product. There are three index two subgroups of $S_{r+2+\frac{k}{2}}\times S_{r+1+\frac{k}{2}}$: namely $A_{r+2+\frac{k}{2}}\times S_{r+1+\frac{k}{2}}$, $S_{r+2+\frac{k}{2}}\times A_{r+1+\frac{k}{2}}$, and $(S_{r+2+\frac{k}{2}}\times S_{r+1+\frac{k}{2}})^{+}$. As there are odd permutations in $P$, odd permutations in $Q$, and odd permutations in $P\diamond Q$, we can rule out all three cases and conclude that $\Gamma\cong S_{r+2+\frac{k}{2}}\times S_{r+1+\frac{k}{2}}$.

∎

Lemma 3.7**.**

For each rank $r\geq 4$ the permutation group $Sm(r)$ given by the following graph is a string C-group isomorphic to $(S_{r}\times S_{r+3})^{+}$.

[TABLE]

Proof.

Let $\Gamma=\langle\rho_{0},\ldots,\rho_{r-1}\rangle=Sm(r)$. We proceed by induction on the rank $r$, where the base case of $r=4$ can be verified using Magma. The group $\Gamma_{0}$ is isomorphic to $Sm(r-1)$, and by induction we may assume that $\Gamma_{0}$ is a string C-group isomorphic to $(S_{r-1}\times S_{r+2})^{+}$. Therefore, it can be seen that $\Gamma\cong(S_{r}\times S_{r+3})^{+}$, and it remains to show that $\Gamma$ is a string C-group. The group $\Gamma_{r-1}$ is isomorphic to the sesqui-extension of a group $Sp(r-1,0)$ through $\rho_{r-3}$. By parts (3) and (4) of Lemma 2.6, we can see that $\Gamma_{r-1}$ is a string C-group. Finally, $\Gamma$ is a string C-group by Lemma 2.1 as $\Gamma_{0,r-1}\cong(S_{r-2}\times S_{r+1}\times S_{2})^{+}$ which is maximal in $\Gamma_{0}$. ∎

Lemma 3.8**.**

For each rank $r\geq 6$ the permutation group $Sy(r,k)$ given by the following graph is a string C-group isomorphic to $(S_{r+\frac{k}{2}}\times S_{r+3+\frac{k}{2}})^{+}$ for all $k\equiv 2\pmod{4}$, with $k\geq 0$, and

[TABLE]

for each rank $r\geq 6$ the permutation group $Sy(r,k)$ given by the following graph is a string C-group isomorphic to $(S_{r+\frac{k}{2}}\times S_{r+3+\frac{k}{2}})^{+}$ for all $k\equiv 0\pmod{4}$, with $k\geq 0$, and

[TABLE]

Proof.

This can be shown by induction on $k$. When $k=0$ this representation is given in Lemma 3.7. Assuming by induction that $Sy(r,k)$ is a string C-group, it follows that $Sy(r,k+2)$ is a string C-group by applying the dual rank reduction of Proposition 2.8 to $Sy(r+1,k)$; in order to apply the dual rank reduction, we need that $r+1\geq 7$. ∎

Lemma 3.9**.**

For each rank $r\geq 6$ the permutation group $L(r,k)$ given by the following graph is a string C-group isomorphic to $S_{r+3+k}$, for all odd $k\geq 0$, and

[TABLE]

for each rank $r\geq 6$ the permutation group $L(r,k)$ given by the following graph a string C-group isomorphic to $S_{r+3+k}$, for all even $k\geq 0$.

[TABLE]

Furthermore $L_{r-1}$ is isomorphic to $S_{r+2+k}$.

To clarify, we note that there are $k+2$ edges of labels either 0 or 1 in each such graph.

Proof.

This can be shown by induction on $k$. When $k=0$ this representation was shown to be a string C-group for all $r\geq 6$ in [8]. Assuming by induction that $L(r,k)$ is a string C-group, it follows that $L(r,k+1)$ is a string C-group by applying the rank reduction of Proposition 2.8 to $L(r+1,k)$. The structure of $L_{r-1}(r,k)$ follows, again using Proposition 2.8, from its relationship to $FL(r-1,0)$ from Lemma 3.1. ∎

Lemma 3.10**.**

For each rank $r\geq 6$ the permutation group $M(r,k)$ given by the following graph is a string C-group, for all $k\equiv 2\pmod{4}$, with $k\geq 0$, and

[TABLE]

for each rank $r\geq 6$ the permutation group $M(r,k)$ given by the following graph is a string C-group, for all $k\equiv 0\pmod{4}$, with $k\geq 0$.

[TABLE]

We note that permutation degree of $M(r,k)$ is $2r+5+k$, and there are $k+4$ edges of labels either 0 or 1 in such a graph.

Proof.

The group $M(r,k)$ is the mix of $L(r,k)$ with $L_{r-1}(r,k)$. Thus $M(r,k)$ is a string C-group by Proposition 2.3. ∎

Lemma 3.11**.**

For each rank $r\geq 6$ the permutation group $Sl(r,k)$ given by the following graph is a string C-group isomorphic to $S_{2r+1+k}$ , for $k\equiv 2\pmod{4}$ with $k\geq 0$.

[TABLE]

We note that there are a total of $\frac{k}{2}+1$ edges of label 0 in such a graph.

Proof.

Let $\Gamma=\langle\rho_{0},\ldots,\rho_{r-1}\rangle=Sl(r,k)$. The group $\Gamma_{0}$ is a string C-group, as it is isomorphic to $Sp(r-1,\frac{k}{2})$ from Lemma 3.6, and thus $\Gamma_{0}$ is isomorphic to $S_{r+1+\frac{k}{2}}\times S_{r+\frac{k}{2}}$. Since $\Gamma_{0}$ is maximal in $S_{2r+1+k}$, we conclude that $\Gamma\cong S_{2r+1+k}$.

Let $Sh(r-1,k)=\langle a_{0},\ldots,a_{r-2}\rangle$, from Lemma 3.3. By part (2) of Lemma 2.6, since $(a_{r-3}a_{r-2})^{3}=1$, we know that $\Gamma_{r-1}\cong Sh(r-1,k)\times S_{2}$, and thus $\Gamma_{r-1}$ is a string C-group isomorphic to $(S_{2}\wr S_{r-1+\frac{k}{2}})^{+}\times S_{2}$. Notice that although $\Gamma_{r-1}$ is not transitive, it still has an imprimitive block structure, with blocks of size two, now with one more block, and also a block of size one. It remains to show that $\Gamma_{r-1}\cap\Gamma_{0}\cong\Gamma_{0,r-1}$ which we will do by analyzing the orbits of $\Gamma_{r-1}\cap\Gamma_{0}$.

If $\alpha\in\Gamma_{r-1}\cap\Gamma_{0}$ then: $\alpha$ preserves the two orbits of $\Gamma_{0}$; $\alpha$ preserves the three orbits of $\Gamma_{r-1}$, namely, it fixes the vertex of the graph which is incident to only an edge of label $r-1$, and it either fixes the vertex of the graph which is incident only to an edge of label $r-2$, or it it interchanges it with the other vertex on that edge of label $r-2$. Finally, $\alpha$ preserves the block structure of $\Gamma_{r-1}$. Thus, we can see $\alpha$ acting only on the blocks, since each block consists of two elements in different $\Gamma_{0}$ orbits. Therefore, $\Gamma_{r-1}\cap\Gamma_{0}\leq(S_{r-1+\frac{k}{2}}\times S_{2})$. It is easy to check that $\Gamma_{0,r-1}\cong(S_{r-1+\frac{k}{2}}\times S_{2})$, and thus $\Gamma$ is a string C-group.

∎

4 Odd degree and high rank

In this section we deal with alternating groups of odd permutation degree, represented as string C-groups of rank at least seven.

Theorem 4.1**.**

For each $r\geq 7$ the permutation group $S(r,k)$ given by the following graph is a string C-group isomorphic to $A_{2r+1+k}$ for all $k\equiv 2\pmod{4}$, with $k\geq 0$.

[TABLE]

Proof.

Let $\Gamma=\langle\rho_{0},\ldots,\rho_{r-1}\rangle=S(r,k)$. The group $\Gamma_{0}\cong Sy(r-1,k)$ is a string C-group isomorphic to $(S_{r-1+\frac{k}{2}}\times S_{r+2+\frac{k}{2}})^{+}$ by Lemma 3.8. The group $\Gamma_{r-1}\cong P\diamond Q$ where $P=Sl(r-1,k)$ is a string C-group isomorphic to $S_{2r-1+k}$, and $Q$ is a single involution extending $\rho_{r-3}$. Since $\Gamma_{r-1}$ only contains even permutations, by parts (3) and (4) of Lemma 2.6, $\Gamma_{r-1}$ is a string C-group isomorphic to $S_{2r-1+k}$. Finally, by Lemma 3.6, the group $\Gamma_{0,r-1}\cong Sp(r-2,k)$ is isomorphic to $S_{r+\frac{k}{2}}\times S_{r-1+\frac{k}{2}}$, which is maximal in $\Gamma_{0}$. Thus $\Gamma$ is a string C-group by Lemma 2.1.

It is clear that $\Gamma$ is a subgroup of $A_{2r+1+k}$. The main theorem of [10] shows that $(S_{r-1+\frac{k}{2}}\times S_{r+2+\frac{k}{2}})^{+}$ is maximal in $A_{2r+1+k}$ and thus $\Gamma\cong A_{2r+1+k}$.

∎

Theorem 4.2**.**

For each $r\geq 7$ the permutation group $B(r,k)$ given by the following graph is a string C-group isomorphic to $A_{2r+3+k}$ for all $k\equiv 2\pmod{4}$, with $k\geq 0$.

[TABLE]

Proof.

Let $\Gamma=\langle\rho_{0},\ldots,\rho_{r-1}\rangle=B(r,k)$.

First let us show that $\Gamma$ isomorphic to $A_{2r+3+k}$. This can be checked using Magma for $B(7,2)$, $B(7,6)$, $B(8,2)$, and $B(9,2)$ where $n=2r+3+k\leq 23$. For all the other cases, $n>24$, and we can use Corollary 1.2 of [11]. In these cases we know that, $|\Gamma|<2^{n}$ or $\Gamma\cong A_{n}$. The group $\Gamma_{0}$ contains a symmetric group acting on one its orbits, and thus $|\Gamma|>(\frac{n+1}{2})!$; however for all $n\geq 25$, $(\frac{n+1}{2})!>2^{n}$, and so $\Gamma\cong A_{n}$.

The group $\Gamma_{0}$ is a string C-group $M(r-1,k)$ from Lemma 3.10. The group $\Gamma_{r-1}$ is a string C-group $Bl(r-1,k)$ by Lemma 3.4. Note that although $\Gamma_{r-1}$ is not transitive, it still has an imprimitive block structure, with blocks of size two, and also a block of size one.

It remains to show that $\Gamma_{r-1}\cap\Gamma_{0}\cong\Gamma_{0,r-1}$ which we will do by analyzing the orbits of $\Gamma_{r-1}\cap\Gamma_{0}$. If $\alpha\in\Gamma_{r-1}\cap\Gamma_{0}$ then: $\alpha$ preserves the two orbits of $\Gamma_{0}$; $\alpha$ preserves the block structure of $\Gamma_{r-1}$, and $\alpha$ preserves the fixed point of $\Gamma_{r-1}$. Thus $\Gamma_{r-1}\cap\Gamma_{0}$ can be seen acting on the $r+1+\frac{k}{2}$ blocks of size two, and is isomorphic to a subgroup of $S_{r+1+\frac{k}{2}}$.

Finally, the group $\Gamma_{0,r-1}\cong P\diamond P$ where $P$ is a string C-group $R(r-2,\frac{k}{2})$ isomorphic to $S_{r+1+\frac{k}{2}}$, by Lemma 3.2, and thus $\Gamma$ is a string C-group.

∎

Corollary 4.3**.**

For each $r\geq 7$, and each $n\geq 2r+1$, there is a string C-group representation of $A_{n}$ of rank $r$.

Proof.

When $n=2r+1$ this follows from Theorem 7.2 of [5]. When $n=2r+3+4j$ for some integer $j$, it follows from Theorem 4.1. Finally, when $n=2r+5+4j$ for some integer $j$, it follows from Theorem 4.2. ∎

5 Even degree and high rank

In this section we deal with alternating groups of even permutation degree, represented as string C-groups of rank at least seven.

Theorem 5.1**.**

For each rank $r\geq 6$ the permutation group $D(r,k)$ given by the following graph is a string C-group isomorphic to $A_{2r+2+k}$, for all $k\equiv 2\pmod{4}$, with $k\geq 0$, and

[TABLE]

for each rank $r\geq 6$ the permutation group $M(r,k)$ given by the following graph is a string C-group isomorphic to $A_{2r+2+k}$, for all $k\equiv 0\pmod{4}$, with $k\geq 0$.

[TABLE]

To clarify this notation we both include an example $D(7,4)$ and remark that there are $k+4$ edges with labels 0 or 1 in each such graph.

Proof.

This is proved by induction on $k$. When $k=0$, $D(r,k)$ was shown to be a string C-group isomorphic to $A_{2r+2}$ in Theorem 7.1 of [5]. To see that $D(r,k+2)$ is a string C-group isomorphic to $A_{2r+2+k+2}$, we notice that you obtain $D(r,k+2)$ from $D(r+1,k)$ using the rank reduction of Proposition 2.8, which is assumed to be a string C-group isomorphic to $A_{2r+2+k+2}$ by induction. We can apply the rank reduction to $D(r+1,k)$ as long as $r+1\geq 7$. ∎

Corollary 5.2**.**

For all $r\geq 7$, the alternating group $A_{n}$ can be represented as a rank $r$- string if $n$ is even and $n\geq 2r+2$.

6 Low Ranks and Extension Construction

In this section we deal with alternating groups represented as rank 4, 5, or 6 string C-groups.

In [14] it was showed that $A_{n}$ is generated by three involutions two of which commute if and only if $n\not\in\{3,4,6,7,8\}$. Additionally, in [15], permutation representations for the string C-groups for each of these $A_{n}$ were provided. Thus it is known exactly which $A_{n}$ are rank 3 string C-groups.

Proposition 6.1**.**

The alternating group $A_{n}$ can be represented as a rank 3 string C-group if and only if $n=5$ or $n\geq 9$.

In order to show which $A_{n}$ can be represented as a rank $r$ string C-group (for $r=4,5,6$) we mainly rely on the following construction, which is similar to one of Pellicer [16] or of Schulte [17].

Definition 6.2**.**

Given an sggi $\Gamma$ whose permutation representation graph $X$ has vertices $[1,\ldots,m]$ and edges labeled $[1,\ldots,r-1]$, we define $\Gamma^{t}$ (for each $t>0$) as the group generated by involutions (with one more generator) whose permutation representation graph has vertices $[1,\ldots,m+t]$ and is obtained by adjoining an alternating path of length $t$ of edges labeled 0,1 to the vertex $m$ of $X$.

As a matter of notation, recall that for an sggi $G=\langle\rho_{0},\ldots,\rho_{r-1}\rangle$ and an integer $k$ we define the groups $G_{\leq k}=\langle\rho_{0},\ldots,\rho_{k}\rangle$.

Proposition 6.3**.**

Let $\Gamma$ be an sggi acting on the points $[1,\ldots,m]$, and for each positive integer $t$ construct $\Gamma^{t}$ as described above. If there is an integer $b\geq 2$ such that $\Gamma^{b}$ is a string C-group, and for each $2\leq k\leq r-2$, the group $\Gamma^{b}_{\leq k}$ acts as a symmetric group on Orbit($\Gamma^{b}_{\leq k},m)$, then for each $t>b$, the group $\Gamma^{t}$ is also a string C-group.

Proof.

Assume that for some $b\geq 2$, $\Gamma^{b}=\langle\rho_{0},\ldots,\rho_{r-1}\rangle$ is a string C-group. Then, by the commuting relationship between the generators, only the elements $\rho_{0}$ and $\rho_{1}$ can have the point $m$ in their support, and thus $m$ is of degree 2 in the natural CPR graph of $\Gamma^{b}$. Let $\Gamma^{t}=\langle\rho_{0},\ldots,\rho_{r-1}\rangle$. To show that $\Gamma^{t}$ is a string C-group we rely on part (b) of Proposition 2E16 of [12]. There is an isomorphism $\phi$ that maps generators of $\Gamma^{t}_{0}$ to generators of $\Gamma^{b}_{0}$, and thus $\Gamma^{t}_{0}$ is also a string C-group. It remains to show that for each $k=0,\ldots,r-2$ that $\Gamma^{t}_{0}\cap\Gamma^{t}_{\leq k}=\langle\rho_{1},\ldots,\rho_{k}\rangle$.

We first deal with small values of $k$. If $k=0$, it is clear by construction that $\rho_{0}\not\in\Gamma^{t}_{0}$ and thus $\Gamma^{t}_{0}\cap\langle\rho_{0}\rangle=\langle 1\rangle$. Let $k=1$ and $\alpha\in\Gamma^{t}_{0}\cap\langle\rho_{0},\rho_{1}\rangle$. We assume, without loss of generality, that $\alpha$ has an even number of factors of $\rho_{1}$ and thus (as it is in $\Gamma^{t}_{0}$) fixes all points greater than or equal to $m$. Since $\alpha\in\langle\rho_{0},\rho_{1}\rangle$ with an even number of factors of $\rho_{1}$ it also fixes all points less than $m$. Therefore $\alpha$ is identity and $\Gamma^{t}_{0}\cap\langle\rho_{0},\rho_{1}\rangle=\langle\rho_{1}\rangle$.

Now assume that $k>2$ and let $\alpha\in\Gamma^{t}_{0}\cap\Gamma^{t}_{\leq k}$ again fixing all points greater than or equal to $m$. Consider $\alpha$ as a word in $\Gamma^{t}_{0}$. Using the isomorphism $\phi$, we get a word $\alpha^{\prime}\in\Gamma^{b}_{0}$ by changing all the factors of $\rho_{j}$ in $\alpha$ to $\rho_{j}$. Under the conditions of the proposition, we assume that the group $\Gamma^{b}_{\leq k}$ acts as a symmetric group on Orbit($\Gamma^{b}_{\leq k},m)$. Additionally, the action of $\alpha$ on all other orbits Orbit($\Gamma^{t}_{\leq k},p)$ is identical to the action of $\alpha^{\prime}$ on Orbit($\Gamma^{b}_{\leq k},p)$, and thus any permutation in $\Gamma^{t}_{\leq k}$ that fixes the points greater than or equal to $m+b+1$ can also be written as a word in $\Gamma^{b}_{\leq k}$. Thus the permutation associated with the word $\alpha^{\prime}$ can also be written as a word in $\Gamma^{b}_{\leq k}$.

Finally, as $\Gamma^{b}$ is a string C-group, $\alpha^{\prime}\in\Gamma^{b}_{\leq k}$ and $\alpha^{\prime}\in\Gamma^{b}_{0}$ implies that $\alpha^{\prime}\in\langle\rho_{1},\ldots,\rho_{k}\rangle$. Again using the isomorphism $\phi$ we get $\alpha\in\langle\rho_{1},\ldots,\rho_{k}\rangle$ as required.

∎

If one can find appropriate groups $\Gamma^{b}$ which fit the conditions of the proposition above, then these can be used to build new string C-groups of the same rank for groups of larger permutation degree.

In particular, for each rank $r\in\{4,5,6\}$, the proof that $A_{n}$ can be represented as a rank $r$ string C-group (for all large $n$) is done by giving examples of string C-groups $\Gamma^{2}$ (see Figures 4, 5, and 6) that satisfy the conditions of Proposition 6.3, and such that for some $j$, the group $\Gamma^{j}$ is isomorphic to $A_{n}$. These examples then yield families of string C-groups $\Gamma^{j+4k}\cong A_{n+4k}$. Thus, for each rank, we need to find an example of $\Gamma^{t}\cong A_{n}$ for each value of $n\mod 4$.

Theorem 6.4**.**

The alternating group $A_{n}$ has a rank 4 string C-group representation if and only if $n=9$, $n=10$, or $n\geq 12$. The alternating group $A_{n}$ has a rank 5 string C-group representation if and only if $n=10$, or $n\geq 12$. Finally, The alternating group $A_{n}$ has a rank 6 string C-group representation if and only if $n=11$ or $n\geq 13$.

Proof.

All of the string C-group representations of $A_{n}$ for $n\leq 10$ were classified in [7], with $n=11,12,13,$ and $14$ subsequently classified in [9]. It remains to show that, for all $n\geq 15$, there are rank $4$, rank $5$, and rank $6$ string C-group representations of $A_{n}$.

For each rank, by listing the generators $\rho_{i}$, we now give four examples of string C-groups $\Gamma^{2}$ that fit the conditions of Proposition 6.3, and show which value of $t$ gives $\Gamma^{t}\cong A_{n}$. To clarify the notation, we include an image of the first family below.

First let us prove that $\Gamma^{t}\cong A_{n}$ for all given families. To do this, we rely on the fact that if $\Gamma^{t}$ is primitive subgroup of $S_{n}$ and $\Gamma^{t}$ contains a 3-cycle then $\Gamma^{t}\geq A_{n}$. All twelve given families of groups $\Gamma^{t}$ are constructed so that for all $t\geq 2$

[TABLE]

is a 3-cycle. Thus we will only need to show why the groups are primitive. All twelve given families of groups $\Gamma^{t}$ are also constructed so that $\langle\rho_{0},\rho_{1},\rho_{2}\rangle$ acts as a symmetric group on the orbit of the point $m$. Therefore, once $k\geq 5$, in all cases $\Gamma^{t}$ will contain a symmetric group acting on more than half its permutation degree, and thus cannot be imprimitive. The isomorphisms between $\Gamma^{t}$ and $A_{n}$ have been found in Magma for $k\leq 4$. Therefore, all the given families of groups yield alternating groups, and using Proposition 6.3, they have been shown to be string C-groups, with the base case of $\Gamma^{2}$ checked in Magma.

∎

We note that for the remaining values of $t$ each of these groups yields a string C-group representation of a symmetric group, as either $\rho_{0}$ or $\rho_{1}$ will be odd, and $\Gamma$ will still contain an alternating group.

7 Main Theorem

In this section, we put together all of our results to summarize the relationship between alternating groups and string C-groups.

Theorem 7.1**.**

For all ranks $r\geq 3$, and all $n\geq 2r+1$, if $n\geq 12$, the alternating group $A_{n}$ has a representation as a rank $r$ string C-group.

Proof.

The small values of $n=12$, $n=13,$ and $n=14$ follow from [9]. Ranks 3, 4, 5, and 6, were considered in Proposition 6.1 and Theorem 6.4. For $r\geq 6$, when $n$ is even this follows from Corollary 5.2, and for $r\geq 7$ when $n$ is odd it follows from Corollary 4.3.

∎

Corollary 7.2**.**

For each rank $r$ it is known exactly which alternating groups can be represented as a string C-group of that rank. Similarly, for each alternating group $A_{n}$, the exact set of ranks of string C-groups for which it can be represented is known.

Proof.

This follows from the previous theorem in addition to Theorem 1.1 of [3], which states that the highest rank of a string C-group representation of $A_{n}$ is $3$ if $n=5$, $4$ if $n=9$, $5$ if $n=10$, $6$ if $n=11$ and $\frac{(n-1)}{2}$ if $n\geq 12$. Moreover, if $n=3,4,6,7$, or $8$, the group $A_{n}$ is not a string C-group.

∎

Corollary 7.3**.**

The group $A_{11}$ is the only alternating group that has string C-group representations of two ranks $r_{1}$ and $r_{2}$, but not all ranks $r_{i}$ in between $r_{1}$ and $r_{2}$.

8 Acknowledgements

The computations in this paper were completed using Magma [1]. I would like to thank Egon Schulte, Barry Monson, Daniel Pelicer, and Gabe Cunningham for insightful conversations that led to various improvements in this paper. I would like to thank Dimitri Leemans and Maria Elisa Fernandes for their support throughout this project. Also I would like to thank Dimitri for first suggesting this problem in 2012; this would not have been possible without them.