ARS MATHEMATICA CONTEMPORANEA
ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 17 (2019) 291-310 https://doi.org/10.26493/1855-3974.1953.c53
(Also available at http://amc-journal.eu)
String C-group representations of alternating groups*
*
Maria Elisa Fernandes
Center for Research and Development in Mathematics and Applications, Department of Mathematics, University ofAveiro, Aveiro, Portugal
Dimitri Leemans
Université Libre de Bruxelles, Département de Mathématique, C.P.216 Algebre et Combinatoire, Bld du Triomphe, 1050 Bruxelles, Belgium
Received 15 March 2019, accepted 15 July 2019, published online 22 October 2019
We prove that for any integer n > 12, and for every r in the interval [3,..., |_n-1 J ], the group An has a string C-group representation of rank r, and hence that the only alternating group whose set of such ranks is not an interval is An.
Keywords: Abstract regular polytopes, Coxeter groups, alternating groups, string C-groups. Math. Subj. Class.: 52B11, 20D06
1 Introduction
String C-group representations have gained much attention in recent years as they are in one-to-one correspondence with abstract regular polytopes. More precisely, given an abstract regular polytope and a base flag of the polytope, one can construct a string C-group representation whose group G is the automorphism group of the polytope that is generated by the set of involutory automorphisms sending the base flag to its adjacent flags [32, Section 2E]. Hence the study of string C-group representations has interest not only for group theory, but also for geometry.
* The authors thank Mark Mixer for observing that there was a mistake somewhere in the case n = 3 (mod 4) in a previous version of this paper. They also thank two anonymous referees for numerous comments that improved a previous version of this paper. This research was supported by the Portuguese Foundation for Science and Technology (FCT - Funda^ao para a Ciencia e a Tecnologia), through CIDMA - Center for Research and Development in Mathematics and Applications, within project UID/MAT/04106/2019 (CIDMA).
E-mail addresses: maria.elisa@ua.pt (Maria Elisa Fernandes), dleemans@ulb.ac.be (Dimitri Leemans)
©® This work is licensed under https://creativecommons.org/licenses/by/4.0/
Abstract
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Classifications of string C-group representations received a big impetus thanks to experimental work of Leemans and Vauthier [31] and also Hartley [20]. These were pushed further for instance in [11, 15, 21, 27]. The results obtained in [31] quickly led to the determination of the highest rank of a string C-group representation of Suzuki groups [26]. Other families of almost simple groups were then investigated: the almost simple groups with socle PSL(2, q) [14, 28, 29], groups PSL(3, q) and PGL(3, q) [5], groups PSL(4, q) [3], small Ree groups [30], orthogonal and symplectic groups in characteristic 2, and finally, symmetric groups [16] and alternating groups [17,18]. In particular, only the last four families gave rise to string C-group representations of arbitrary large rank. In [2], it is shown that, for all integers m > 2, and all integers k > 2, the orthogonal groups O± (2m, F2k) act on abstract regular polytopes of rank 2m, and the symplectic groups Sp(2m, F2k) act on abstract regular polytopes of rank 2m + 1. A symmetric group Sn is known to have string C-group representations of highest rank n - 1 [6] and an alternating group An is known to have string C-group representations of highest rank |_n—1J when n > 12 [8]. It is worth noting that not only almost simple groups have been investigated. For instance, Cameron, Fernandes, Leemans and Mixer determined the maximal rank of a string C-group representation of a transitive permutation group in [7]. Conder determined in [9] the smallest string C-group representations of rank r. It turns out that when r is at least 9, all such groups are 2-groups. Further studies on string C-group representations of 2-groups are available for instance in [23, 24].
The authors looked at the symmetric groups in [16] and proved three important facts. Firstly, when n > 5, the (n - 1)-simplex is, up to isomorphism, the unique string C-group representation of Sn with rank n - 1. Secondly, they showed that when n > 7, there is also, up to isomorphism, a unique string C-group representation of rank n - 2. And finally, they showed that for every n > 4, and for every integer r in the interval [3,..., n - 1], a symmetric group Sn has at least one string C-group representation of rank r. Therefore, the symmetric groups have no gaps in their set of ranks. The first and second theorems have been extended in [19] where the authors of this paper, together with Mark Mixer, classified string C-group representations of rank n - 3 (for n > 9) and n - 4 (for n > 11) of the symmetric group Sn.
Also with Mixer, the authors produced in [17, 18] string C-group representations of rank |(n - 1)/2j of the alternating groups, with n > 12. In the process of obtaining these results, they computed all string C-group representations of An with n < 12. They found that the set of ranks for the alternating groups of small degree were as given in Table 1. The
Table 1: Set of ranks for small alternating groups.
Group	Set of ranks
A5	{3}
Ae	0
A7	0
As	0
Aq	{3,4}
A10	{3, 4, 5}
An	{3, 6}
A12	{3, 4, 5}
M. E. Fernandes and D. Leemans: String C-group representations of alternating groups 293
case n = 11 turned out to be special in the sense that it was the only example encountered so far of a group whose set of ranks presented gaps. In this paper, we prove a similar result as the third theorem of [16]. Our main result is stated as follows.
Theorem 1.1. For n > 12 and for every 3 < r < |_(n — 1)/2_|, the group An has at least one string C-group representation of rank r.
This theorem shows indeed that the case n = 11 is special among the alternating groups. The main tool in the proof of our main theorem is to find good permutation representation graphs that turn out to be CPR graphs, for every rank 3 < r < |(n — 1)/2j once n is fixed. We use a proof similar to that of the third theorem of [16] to tackle most cases and are just left dealing with finding string C-group representations of ranks four and five for An when n is even, and ranks four, five and six, when n = 3 (mod 4).
The paper is organised as follows. In Section 2, we recall the basic definitions about string C-groups. In Section 3, we recall the definitions of permutation representation graphs and CPR-graphs and give some results that will be useful in proving Theorem 1.1. In Section 4, we prove Theorem 1.1. In Section 5, we give some final remarks.
As to notation for groups, we denote a cyclic group of order n by Cn, a dihedral group of degree n and order 2n by Dn, and by pn an elementary abelian group of order pn. Also, if G is a permutation group, the group G+ is the subgroup of G generated by the even permutations in G, and if G+ = G (so that all elements of G are even) then we call G an even permutation group.
2 String C-groups
An abstract polytope is a combinatorial object which generalizes a classical convex poly-tope in Euclidean space. When the automorphism group of an abstract polytope acts regularly on its set of flags, the polytope is called regular, and in that case, its automorphism group admits a string C-group representation. Additionally, each abstract regular polytope can be constructed from a string C-group representation, and thus abstract regular poly-topes and string C-groups representations are basically the same objects. For more details on the subject see [32, Section 2E].
A Coxeter group is a group with generators p0,..., pr-1 and presentation
{pi I (.PiPj)mi'j = £ for all i, j e {0,..., r — 1})
where £ is the identity element of the group, each mi j is a positive integer or infinity, mi,i = 1, and mijj = mjii > 1 for i = j. It follows from the definition, that a Coxeter group satisfies the next condition called the intersection property.
VJ,K c{0,...,r — 1}, {pj | j e J)n{pfc | k e K) = {pj | j e J n K)
A Coxeter group G can be represented by a Coxeter diagram D. This Coxeter diagram D is a labelled graph which represents the set of relations of G. More precisely, the vertices of the graph correspond to the generators pi of G, and for each i and j, an edge with label mi j joins the ith and the jth vertices; conventionally, edges of label 2 are omitted. By a string (Coxeter) diagram we mean a Coxeter diagram with each connected component linear. A Coxeter group with a string diagram is called a string Coxeter group.
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More generally, we define a string group generated by involutions, or sggi for short, as a pair (G, S) where G is a group, S := {p0,..., pr-1} is a finite set of involutions of G that generate G and that satisfy the following property, called the commuting property.
Vi, j e {0,... ,r - 1}, |i - j| > 1 ^ (pipj)2 = 1
Finally, a string C-group representation of a group G is a pair (G, S) that is a sggi and that satisfies the intersection property. In this case the underlying "Coxeter" diagram for (G, S) is a string diagram. The (Schlafli) type of (G, S) is {p1,... ,pr-1} where p is the order of pi-1pi, i e {1,..., r - 1}, and the rank of a string C-group representation (or of a sggi) (G, S) is the size of S. When the context is clear, we sometimes do not specify the set of generators S and we talk about a string C-group G instead of a string C-group representation (G, S).
The set of ranks of a group G is the largest set of integers I such that for each r e I, there exists at least one string C-group representation of G with rank r.
Let r := (G, S) be a sggi with S := {p0,..., pr-1}. We denote by G/ with I C {0,..., r - 1} the subgroup of G generated by the involutions with indices that are not in I and let r/ := (G/, {pj : j e I}); it follows from the definition that if r is a string C-group representation of G, each r/ is itself a string C-group representation of G/. Also, for i, j e {0,..., r - 1}, we denote Gj = (pj | j = i) and Gijj := (Gjj. The following two results show that when r0 and rr-1 are string C-group representations, the intersection property for (G, S) is verified by checking only one condition.
Proposition 2.1 ([32, Proposition 2E16]). Let r := (G, S) be a sggi with S := {p0,..., pr-1}. Suppose that r0 and rr-1 are string C-group representations. If G0 n Gr-1 = G0,r-1, then r is a string C-group representation of G.
We point out that the inclusion G0 n Gr-1 > G0,r-1 is immediate, and thus we only need to check that G0 n Gr-1 < G0,r-1. The following proposition makes it even simpler to check if a pair (G, S) is a string C-group representation when G0,r-1 is a maximal subgroup of either G0 or Gr-1 (or both).
Proposition 2.2 ([18, Lemma 2.2]). Let r = (G, S) be a sggi with S := {p0,..., pr-1} and G := (S). Suppose that r0 and rr-1, are string C-group representations of G0 and Gr-1 respectively. If pr-1 e Gr-1 and G0,r-1 is maximal in G0, then r is a string C-group representation of G.
3 Permutation representation graphs and CPR graphs
Let G be a group of permutations acting on a set {1,..., n}. Let S := {p0,..., pr-1} be a set of r involutions of G that generate G. We define the permutation representation graph G of G, as the r-edge-labeled multigraph with n vertices and with an i-edge {a, b} whenever apj = b with a = b.
The pair (G, S) is a sggi if and only if G satisfies the following properties:
1.	The graph induced by edges of label i is a matching;
2.	Each connected component of the graph induced by edges of labels i and j, for |i—j | > 2, is a single vertex, a single edge, a double edge, or a square with alternating labels.
M. E. Fernandes and D. Leemans: String C-group representations of alternating groups 295
When (G, S) is a string C-group representation, the permutation representation graph G is called a CPR graph, as defined in [33]. In rank 3, there are a couple of known results to determine if a 3-edge-labeled multigraph is a CPR graph. For higher ranks, no such arguments were accomplished.
One simple example of a CPR graph is the one corresponding to the (n - 1)-simplex as follows:
O-
-O
-o
-o
-O"
n — 2	n— 1
■o-o-o
In [16], for each rank 3 < r < n — 2, a string C-group representation of rank r of Sn was found. In [18], the authors constructed a string C-group representation of rank r > 4 of An for some n. This is summarized in the following two theorems, and the associated CPR graphs are given in Table 2.
Theorem 3.1 ([16, Theorem 3]). For n > 5 and 3 < r < n — 2, there is a string C-group representation of rank r and type {n — r + 2,6, 3r-3} of Sn.
Theorem 3.2 ([18, Theorem 1.1]). For each rank k > 3, there is a string C-group representation of rank k of An for some n. In particular, for each even rank r > 4, there is a string C-group representation of A2r+1 of type {10,3r-2}, and for each odd rank q > 5, there is a string C-group representation of A2q+3 of type {10,3q-4,6,4}.
Table 2: String C-group representations of Sn and An
Group	Schlafli type	CPR graph
Sn (3 < r < n - 2)	{n - r + 2, 6, 3r-3}	O^O^O q0Q!Q2Q3Q Or-^r-1o
A2r+1 (r even and > 4)	{10, 3r-2}	12 3 r—2 r—1 o—o^p—9......(>-<>—o 0 0 0 0 0 0 ......°r—A—<^
A2r+3 (r odd and > 5)	{10, 3r-4, 6,4}	12 3 r—2 r—1 r—2 o—O^p—O......o—o—o—o 0 0 0 0 0 0 0 ......or^—^r—o
0
1
2
3
Permutation representation graphs are a very useful tool for the construction of string groups generated by involutions. We will use them in the proof of our main theorem.
The term sesqui-extension was first introduced in [18]. Let us recall its meaning. Let $ = (a0,..., ad~1) be a sggi, and let t be an involution in a supergroup of $ such that t e $ and t centralizes $. For fixed k, we define the group $* = («¿Tni | i e {0,..., d — 1}) where ni = 1 if i = k and 0 otherwise, and call this the sesqui-extension of $ with respect to ak and t. In particular, a permutation representation graph having two connected components, one of which is a single k-edge and the other contains at least one k-edge, represents a sesqui-extention of a group (the group corresponding to the biggest component) with respect to the generator k.
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Proposition 3.3 ([17, Proposition 5.4]). If $ = (a | i = 0,..., d — 1} and
$* = (o^ | i € {0, .. ., d - 1}}
is a sesqui-extension of $ with respect to ak, then ($, {a | i = 0,..., d — 1}) is a string C-group representation if and only if ($*, {«.¿t^ | i € {0,...,d — 1}}) is a string C-group representation. Moreover one of the following situations occur.
(1)	t € $*, in which case $* is isomorphic to $ x (t} = $ x C2; or
(2)	t € $*, in which case $* is isomorphic to $.
Sesqui-extensions will be used later to check the intersection condition on the permutation representations of the groups of our main theorem.
We also apply the techniques used in the proof of Theorem 3.1 based on a construction of Hartley and Leemans available in [22]. The key of the proof of Theorem 3.1 was to start from the CPR graph of the (n — 1)-simplex with generators pi,..., pn-1 where p is the transposition (i, i + 1) in Sn. Let d = n — 1. At each step, we start with a string C-group representation of rank d and generators p1,..., pd. We replace pd-2 by pd-2pd and we drop pd. As proved in [16], we get in this way a new string C-group representation with generators p1,..., pd-1. We can repeat this until d =3. We give in Table 3 an example of this process for S7.
Table 3: The induction process used on S7.
Generators	CPR graph	Schlafli type
(1, 2), (2, 3), (3, 4), (4, 5), (5, 6), (6, 7)		{3, 3, 3, 3, 3}
(1, 2), (2, 3), (3, 4), (4, 5)(6, 7), (5, 6)	Q 1Q2Q^Q4Q^Q4Q	{3, 3, 6,4}
(1, 2), (2, 3), (3,4)(5, 6), (4, 5)(6, 7)	q 1q2Q^Q4Q^Q4Q	{3, 6, 5}
(1, 2), (2, 3)(4, 5)(6, 7), (3, 4)(5, 6)		{6, 6}
In order to prove that the permutation groups of our main theorem are isomorphic to alternating groups we use the following results.
Theorem 3.4 ([25]). Let G be a primitive permutation group of finite degree n, containing a cycle of prime length fixing at least three points. Then G > An.
Proposition 3.5 ([17, Proposition 3.3]). Let G = (po,..., pr-1) be a transitive permutation group acting on the points {1,... ,n} with n > 5, andlet G* = (p0,... ,pr-1,pr , pr+1), where
pr = (i, n + 1)(n + 2, n + 3) for some i G {1,..., n} pr+1 = (n + 1, n + 2)(n + 3, n + 4).
Then G* = An+4 or Sn+4, depending on whether or not G is even.
M. E. Fernandes and D. Leemans: String C-group representations of alternating groups 297
Proposition 3.6. The following graph, with n > 8 vertices, n even and r G {3,..., }, is a CPR graph for (S n— x S n+4)+.
..................o^-V^o
..................
Proof. Let r := (G, S) be the sggi having the permutation representation given by the
graph of this proposition. Let us first consider r = 3.
.........
.........
We see that r0 and r2 are string C-group representations and as G0 n G2 = G0,2 = C2, r is itself a string C-group representation by Proposition 2.1.
Let us prove that G is isomorphic to (S n— x S n+4 ) . We first prove that G contains the
v 2	2 '
3-cycles (l, 2,3) and (4,5,6) (the vertices of the above graph on the right). Let l be the least integer such that (popi)' fixes all the vertices of the component of the graph on the bottom. We see that (p1p2)2 = (l, 2,3)(4,5, 6). The latter element conjugated by (p0p1)' is equal to a = (a, 6, c)(4, 5, 6) with {a, 6, c} n {l, 2, 3} = {l}. Hence (a(p1p2)2)5 = (4, 6, 5) and (l, 2, 3) = (4,6, 5)(pip2)2.
Now by transitivity in each of the two components of the graph we find that G has a subgroup isomorphic to A n— x A n+4. As in addition p2 G A n—; x A n+4 and G is a
22	22
group of even permutations, the group G is isomorphic to (S n— x S n+4 ) + .
Now let r > 3. We may assume by induction that rr-1 is a string C-group representation and Gr_ 1 is isomorphic to (Sn— x Sn+2) +. In addition r0 is a string C-group
v 2	2 '
representation with group G0 isomorphic to Sr-1. By the intersection of the orbits of G0 and Gr-1 we conclude that G0 n Gr-1 and G0,r-1 are both isomorphic to Sr_2. Therefore r is a string C-group representation of G. Moreover it is clear that G is isomorphic to
(S n-4 x S n+4 ) .	n
Proposition 3.7. The following graph, with n > l0 vertices, n even and r G {5,..., n-2 }, is a CPR graph for Sn.
çy^y^y^y^y^y^..................O^-V^O
r-2
O—:—O.........O——O-^^O.........O——-o
0	1	0	12	r — 2 r — 1
Proof. Let r := (G, S) be the sggi having the permutation representation given by the graph of this proposition. The permutation representation graph is connected, hence G is transitive. Let x be the first point on the left of the graph. The stabilizer of x has at most
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the same orbits as G0. Consider the vertices y and z as in the following graph.
..................OT-V-io
r —2
O——O..........o—0—o——o—2—O..........o—^—tO
0	1	0	12	r-2 r — 1
We see that ypplP0 = z and p2lP0 fixes x. More generally the appropriate conjugations of p2 by powers of p0pT fuse the orbits of G0 while fixing x. Hence G is 2-transitive and therefore primitive. Moreover, it contains a 3-cycle (explicitly given in the proof of Proposition 3.6) and an odd permutation. Hence, by Theorem 3.4, it is isomorphic to Sn—1. By Proposition 3.3 and [17, Table 2] we may conclude that r0 is a string C-group representation of the group C2 x (C2 i Sr—t). By Proposition 3.6, the sggi rr—t is a string C-group representation of (S n— x S n+2 ) +. From the intersection of the orbits of G0 and Gr—1 we also conclude that G0 n Gr—1 = G0,r—1 = C2 x (Sn— x Sn+i ) + . Hence r is a string C-group representation.	□
Proposition 3.8. The following graph, with n > 10 vertices, n even and r G {3,..., }, is a CPR graph for (S n-4 x S n±i)+.
..................O^V^O
..................
Proof. Similar to that of Proposition 3.6.	□
Proposition 3.9. The following graph, with n > 12 vertices, n even and r G {5,..., }, is a CPR graph for Sn.
o-^—-^o..................O^V—^P
r-2
o—-—O.........O——O-^^O.........o——TO
1	0	12	r —2 r —1
Proof. Similar to that of Proposition 3.7.	□
Proposition 3.10. The following graph, with n > 8 vertices, n even and r = n/2, is a CPR graph for Sn.
^ 0 ^ 1 ^ 2 ^	r—2r—l
r —2
Proof. Let r := (G, S) be the sggi having the permutation representation given by the graph of this proposition. Removing the 0-edge from the graph we get a CPR graph for a symmetric group of degree n - 1 (see Table 2 of [17]). Hence r0 is a string C-group
M. E. Fernandes and D. Leemans: String C-group representations of alternating groups 299
representation. Now consider the sggi $ := (H, T) with the following permutation representation graph.
For r = 4, $ is a string C-group representation with H isomorphic to C2 x S4. Assume by induction that $r-2 is a string C-group representation with Hr-2 isomorphic to Sr-1 x Sr-3. As $0 is a string C-group representation and H0 n Hr-2 < Sr-2 x Sr-3 = H0,r-2, $ is a string C-group representation. Moreover H is isomorphic to Sr-1 x Sr-3. Now by Proposition 3.3 the sggi rr-1 is a string C-group representation and Gr-1 is isomorphic to C2 x Sr-1 x Sr-3. By the intersection of the orbits of G0 and Gr-1 we find that G0 n Gr-1 = G0,r-1 Hence r is a string C-group representation. As G0 is isomorphic to Sn-1 and stabilizes the first vertex on the left, we conclude that G is isomorphic to Sn. □
Proposition 3.11. The following graph with n vertices, n = 3 (mod 4) and n > 11, is a CPR graph for Sn.
0
0
3
Proof. Let r := (G, S) be the sggi having the permutation representation given by the graph of this proposition. The group G3 is an even transitive group containing a 3-cycle, namely (pip2 )4, and the stabilizer of a point in G3 is transitive on the remaining points. Hence by Theorem 3.4 the group G3 is isomorphic to An-1. Consequently G is isomorphic to Sn. Moreover as G3 is a simple group generated by three independent involution, the sggi r3 is string C-group representation. It is also easy to check that r0 is string C-group representation and that G3 n G0 = G0,3, as it is sufficient to consider the case n =11. Hence r is a string C-group representation and G is isomorphic to Sn as wanted. □
4 Proof of Theorem 1.1
For each n > 12, the group An has at least one string C-group representation of rank three. Indeed, we can rely on [12, 13] which covers all but a small number of small cases that can be easily dealt with Magma [1], or [34]. Hence we have to construct examples of rank 4 and above. Also, the case where n = 12 is done in [18], hence we may assume n > 12.
We divide the rest of the proof is a series of theorems depending on the values of n and r as described in Table 4. Theorem 4.1 comes from [17], and we use it in Theorem 4.2 to construct string C-group representations of rank 6 < r < (n - 2)/2 for n even.
4.1 The even case
We will construct a family of CPR graphs of even ranks "reducing" the rank of a CPR graph having highest possible rank. Let us consider the graph given in the following theorem.
Theorem 4.1 ([17]). If n > 14 is even and r = n—2 > 6, then the following graph is a
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CPR graph for An.
o—o—o—o—o—o—o-
_ r —3 _ r — 2 r — 1
0—O-0—o
r-3
.........°r-3~ r2~ r 1
rO—rO
r-3
O
Moreover the corresponding string C-group representation has type {5, 6, 3r 6, 6,6,3}.
Theorem 4.2. If n is an even integer, n > 14 and 6 < r < n22, then the group An admits a string C-group representation of rank r, with Schlafli type {lcm(4 + i, i), 6,3r-6,6,6,3} where i = (n — 2)/2 — r + 1, and with the following CPR graph
r 3 r 2 r 1
r—3 r—3
r 3 r 2 r 1
for (n = 2 (mod 4) and n — r even) or(n = 0 (mod 4) and n — r odd) and the following CPR-graph
r-3 r-2 r-1
r—3 r— 3
r 3 r 2 r 1
for (n = 2 (mod 4) and n — r odd) or (n = 0 (mod 4) and n — r even).
Proof. From the graph of Theorem 4.1 we construct a family of graphs with n vertices and r G {6,..., n-2} adding, on the top and on the bottom of the graph, two sequences
0
1
0
1
2
3
1
0
1
2
3
Table 4: The structure of the proof depending on n and r.
n	r	Reference
n even	6 < r < (n - 2)/2	Theorem 4.2
n = 0 (mod 4)	r = 5 r = 4	Theorem 4.6 Theorem 4.5
n = 2 (mod 4)	r = 5 r = 4	Theorem 4.4 Theorem 4.3
n = 1 (mod 4)	4 < r < (n - 1)/2	Theorem 4.7
n = 3 (mod 4)	r = (n - 1)/2 7	< r < (n - 1)/2 and r odd r = (n - 1)/2 - 1 8	< r < (n - 1)/2 and r even r = 4 r = 5 r = 6	Theorem 4.8 Theorem 4.9 Theorem 4.10 Theorem 4.11 Theorem 4.12 Theorems 4.13 and 4.15 Theorem 4.14
M. E. Fernandes and D. Leemans: String C-group representations of alternating groups 301
of edges, of the same size, with alternate labels 0 and 1. So we have the following two possibilities.
0„1„0„1„0„1
0 1
10101
0 „ 1 „ 2 „ 3
0 12 3
r —3 „ r — 2 r — 1
r—3
r-3
r-3 r —2 r—1
r-3
0 12 3	r-3 r—2 r —1
0 1 2 3	„r—3„ r —2 _ r—1
r—3
Let r := (G, S) be the sggi having the permutation representation graph above. The statement holds for n =14 and r = 6 by Theorem 4.1. Assume n > 14.
The involution p1 can be decomposed as p1 = t«i where a is the restriction of p1 to the biggest G0-orbit and t is the restriction of p1 to the union of G0-orbits of size 2. The following CPR graph has group isomorphic to (2r : Sr)+ as shown in [17, Lemma 6.6]. It is exactly the graph we obtain by replacing p1 by a1 and forgetting about the points fixed by Go.
-3 r-2 r-1
-3 r-3
23
-3 r-2 r-1
We find that a1 = p2p1p2p1p2 G G0, then also t g G0 and therefore by Proposition 3.3, G0 is a sesqui-extension of the group (2r : Sr)+ and G0 is isomorphic to C2 x (2r : Sr )+ = 2r : Sr as t g G0. Moreover, r0 is a string C-group representation.
We use a similar argument to prove that rr—1 is a string C-group, starting from the CPR graph given in Proposition 3.7 when (n = 2 (mod 4) and n — r even) or (n = 0 (mod 4) and n—r odd), and from the CPR graph given in Proposition 3.9 when (n = 2 (mod 4) and n — r odd) or (n = 0 (mod 4) and n — r even). In that case, however, since the restriction of pr—2pr—3 to the biggest orbit of Gr—1 is an element of even order, Gr—1 = Sn—2. Since An acts primitively on the set of unordered pairs of points, the stabilizer in An of a fixed pair is maximal in An, and such stabilizers have precisely the structure of Gr—1. As Gr—1 is a maximal subgroup of An and pr—1 G Gr—1, it follows that G is isomorphic to An. Let us now prove that G0,r—1 = G0 n Gr—1. The orbits of G0 n Gr—1 have to be suborbits of G0 and of Gr—1, hence G0 n Gr—1 < (C2 x (2r—1 : Sr—1) x C2 )+ = G0,r—1. Hence, by Proposition 2.1, r is a string C-group representation of An.
Let i = (n — 2)/2 — r +1. Then it is easy to see from the CPR-graph that the Schlafli type of the string C-group representation of An of rank r obtained by this construction is {lcm(4 + i, i), 6, 3r—6, 6, 6,3}. The first entry of the symbol comes from the fact that there are 0-1-components on the upper side of the graph and on the lower side of the graph and the upper one has 4 more vertices than the lower one.	□
1
r
It remains to construct examples in rank 4 and 5 for n even. We split the discussion in two cases, namely the case where n = 0 (mod 4) and the case where n = 2 (mod 4).
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Theorem 4.3. If n = 2 (mod 4) with n > 10, then the group An admits a string C-group representation of rank 4, with Schlafli type {5,6, n — 4}, with the following CPR-graph.
(Fi)
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. In this case G3 is a sesqui-extension of a string C-group representation of A5, hence by Proposition 3.3, G3 = C2 x A5 and r3 is a string C-group representation of rank 3. Moreover, G0,3 is isomorphic to C2 x D3 = D6 and therefore G0,3 is maximal in G3. So, by Proposition 2.2, it remains to prove that r0 is also a string C-group representation. Now, r0 3 and r0 i are obviously string C-group representations of dihedral groups. The group G0,i,3 is a cyclic group of order 2 and the subgroups G0,3 and G0,i will have the same intersection no matter what the value of n is. We can thus assume n =10 and check by hand or using Magma that G0 n G3 = G0 3. Hence r0 is a string C-group representation. This concludes the proof that a sggi with permutation representation graph (F^ is a string C-group representation. It remains to show that the four generators generate An. The element p0pi is a 5-cycle and G is primitive, as for instance p0 cannot preserve any block system. Hence, by Theorem 3.4, G is isomorphic to An.
The Schlafli type is obvious from the permutation representation graph.	□
Theorem 4.4. If n = 2 (mod 4) with n > 10, then the group An admits a string C-group representation of rank 5, with Schlafli type {5,5,6, n — 5}, with the following CPR-graph.
(F2)
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. In this case, G4 is a sesqui-extension of a group isomorphic to (S7 x C2)+ = S7 whose CPR graph is given in Table 2 of [17]. Hence r4 is a string C-group representation. By Proposition 3.5 the group G0 is isomorphic to An_i. The subgroup G0,4 is isomorphic to S6, in addition G0,1,4 = D6 and G0,1 = Sn_4. Increasing n will not change the intersection between G01 and G0 4. Hence we can check with Magma that G01 n G0 4 = G0,1,4 for n = 10. Thus r0,1 is a string C-group representation and so is r0 and so is r, as G0 = An-1 and G is transitive. Moreover G is isomorphic to An since it is transitive on n points and the stabilizer of a point in G contains G0 = An-1.
The Schlafli type is obvious from the permutation representation graph.	□
Theorem 4.5. If n = 0 (mod 4) with n > 16, then the group An admits a string C-group representation of rank 4, with Schlafli type {3,12, lcm(n — 8,6)}, with the following CPR-graph.
32
(F3)
32
0
3
2
3
3
3
2
0
3
4
3
4
3
4
3
4
M. E. Fernandes and D. Leemans: String C-group representations of alternating groups 303
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. In this case, G3 is isomorphic to 22 : S3 x S3 and G0,3 is isomorphic to D12 no matter what the value of n is, thanks to the shape of the graph. Observe that the left connected component of the graph, obtained when removing the 3-edges, gives the CPR graph of the octahedron. Thus it can easily be checked with Magma that r3 is a string C-group representation with type {3,12}. The group G0 is transitive on n - 1 points, namely all vertices of the graph except l. Moreover, the stabilizer of l and p in G has at most two more orbits thanks to the connected components of the permutation representation graph obtained by removing edges labelled 0 and 1. The element (pip2p3p2)3 moves point i to point d while fixing both l and p. Hence G0 is 2-transitive on n - 1 vertices (all but l). Therefore G0 is primitive on these points. Now the element (p1p2p3p2) = (l)(p, j, m)(i, e, g, d, h)(a, c, f, b)... has the property that the cycles we did not write are transpositions. Indeed, p1 does not do anything on these points and so the action on these points is given by p2p3p2 = p32 which is an involution. Hence (p1p2p3p2)12 € G0 is a 5-cycle fixing more than three points. By Theorem 3.4, we can therefore conclude that G0 is isomorphic to An-1. As G0 is a simple group, since it is generated by three involutions (namely p1, p2, p3), two of which commute, r0 is a string C-group representation by [10, Theorem 4.1]. It remains to check that G0,3 = G0 n G3 to prove that these graphs give indeed string C-group representations. This can be checked with Magma for n = 12 and the result can be extended for any n.
The Schlafli type is obvious from the permutation representation graph.	□
Theorem 4.6. If n = 0 (mod 4) with n > 12, then the group An admits a string C-group representation of rank 5, with Schlafli type {3,4,6, n — 7}, with the following CPR-graph.
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. In this case, G4 is a sesqui-extension of the group of a string C-group representation of S9, that can be found for instance in the atlas [31]. The sggi r01 is a string C-group representation of Sn-6 and G0,4 is isomorphic to S5 x D4. Now p2p3 has order 6, so G014 is isomorphic to D6 and it is obvious from the permutation representation graph that G0,4 n G0,1 = G0,1,4 and G0,4 n G1,4 = G0,1,4. Hence r0 and r4 are string C-group representations by Proposition 2.1. As G0 n G4 must have orbits that are suborbits of those of G0 and of those of G4, we readily see that G0 n G4 = G0,4. This concludes the proof that every graph of shape (F4) gives a string C-group representation. As G is a primitive group generated by even permutations and (p2p3)2 is a 3-cycle, we see that G is isomorphic to An by Theorem 3.4.
The Schlafli type is obvious from the permutation representation graph.	□
4.2 The odd case
Theorem 4.7. Ifn and r are integers with n > 13, n = 1 (mod 4) and 4 < r < (n —1)/2, then the group An admits a string C-group representation of rank r, with Schlafli type {10, 3^-2} when r = and {10,3r-4,6, — r + 3} when r < n-1, and with the
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following CPR graph.
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. Clearly G is a group of even permutations and it must be primitive as p0 cannot preserve a non-trivial block system. Let us prove that G is isomorphic to An. We see that (p0pi)2 is a 5-cycle, hence by Theorem 3.4, the group G is isomorphic to An. It remains to prove that r satisfies the intersection property. We know that for n =13, the sggi r is a string C-group representation of rank 6 and Schlafli type {10,3,3,3, 3}. It can be checked with Magma that r is also a string C-group representation for n =13 and r G {4,5}. By induction we may assume that Gr-1 is a sesqui-extension of the group of a string C-group representation. Hence by Proposition 3.3, the sggi rr-1 satisfies the intersection property. By the first line of Table 2, it is easy to see that r0 is a string C-group representation. Finally, G0,r-1 = G0 n Gr-1 = Sr-1 x C2. By Proposition 2.1, we conclude that r is a string C-group representation. Using this technique, we have just constructed string C-group representations of rank r for every 4 < r < . Their Schlafli types are {10,32} when r = ^ and {10, 3r-4, 6, ^ - r + 3} when r< ^.	□
The following theorem gives the string C-group representations of rank r = (n - 1)/2 in the case where n = 3 (mod 4).
Theorem 4.8 ([17]). If n and r are integers with n > 15, n = 3 (mod 4) and r = (n -1)/2, then the group An admits a string C-group representation of rank r, with Schlafli type {5,5, 6, 3r-7, 6,6,3}, and with the following CPR graph.
r—3 r — 2 r — 1
-3 r-3
r 3 r 2 r 1
0
3
r
3
From these examples, we construct examples of the same rank but for groups of degree n + 4k where k is an integer, by adding a sequence of alternating 0- and 1-edges of length 4k between the first and the second 2-edge (counting from the left).
Theorem 4.9. If n and r are integers with n > 15, n = 3 (mod 4) and 7 < r < (n-1)/2, r odd, then the group An admits a string C-group representation of rank r, with Schlafli type {n — 2(r — 2), 12, 6, 3r-7, 6, 6, 3}, and with the following CPR graph.
r 3 r 2 r 1
-3 r-3
r 3 r 2 r 1
0
0
2
3
r
3
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. The group G0 is acting as S2(r-1) on the orbit of size 2(r - 1) and as D4 on the orbit of size 4, making G0 isomorphic to A2(r-1) : D4. Observe that G0 has a structure that only depends on the rank, not on the degree of G.
M. E. Fernandes and D. Leemans: String C-group representations of alternating groups 305
The group G0,r-i is isomorphic to S2(r_2) : D4. It is a maximal subgroup of G0. Hence G0 n Gr-1 = G0,r-1.
Let us now prove that r0 and rr-1 are string C-group representations. We start with r0. The group G01 is the same (up to removing the fixed points) as the one of Theorem 4.8. Hence r0 is a string C-group representation. The sggi r0,r-1 has the following permutation representation graph, where there might be more than one 1-edge disconnected from the rest of the graph.
O
r-3 r-2
0-3-0--
•-3 r-3
-3 r-2
If we prove that the sggi corresponding to the following permutation representation graph is a string C-group representation, we may then apply Proposition 3.3 in order to show that r0,r_i is also a string C-group representation.
r-3 r-2
r-3
r-3 r-2
Let us call $ := (H, T) the sggi having this permutation representation graph. By Proposition 3.10 the connected component on the right of the graph above gives a string C-group representation. By Proposition 3.3 the graph that we obtain from the graph pictured above by removing the 2-edge on the left is a CPR graph. Since removing the 2-edge on the left does not change the order of the group H1, by [32, Proposition 2E17] we find that $ is a string C-group representation. Hence r0 is a string C-group representation.
Let us now prove that rr-1 is a string C-group representation.
The group Gr-2,r-1 is a sesqui-extension of the group K of the sggi ^ := (K, U) having the following permutation representation graph.
r-3
o-p
r-3
3
r
2
0
0
2
3
3
Let a and b be the sizes of the connected components of the graph above. For r = 6, K is a sesqui-extension of the group of the string C-group representation of Proposition 3.11, hence by Proposition 3.3, K is isomorphic to Sa = (Sa x 2)+. By induction we may assume that ^r-3 is a string C-group representation and Kr-3 is isomorphic to (Sa-1 x Sb-1) + . As is a string C-group representation and K0 n Kr-3 = K0,r-3 we find that ^ is itself a string C-group representation. Moreover K is clearly isomorphic to (Sa x Sb)+. With this, using Proposition 3.3, we see that rr-2 r-1 is a string C-group representation. Finally G0,r-1 n Gr-2,r-1 < (D4 x S2(r-3) X 2)+ = G0,r-2,r-1.
Hence we have proved that rr-1 is a string C-group representation and therefore G itself is a string C-group.
It is easy to see from the permutation representation graph in the theorem that the Schlafli type of the string C-group representation of rank r of An obtained by this construction is {n - 2(r - 2), 12, 6, 3r-7, 6, 6, 3}.	□
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The previous two theorems enable us to construct examples of all possible odd ranks at least 7 for An with n = 3 (mod 4) and n > 15. We now construct an example of rank (n - 3)/2 for An from the example of rank (n - 1)/2, that we will use to construct all examples of even rank at least 8.
Theorem 4.10. If n and r are integers are such that n > 19, n = 3 (mod 4) and r = (n — 1)/2 — 1, then the group An admits a string C-group representation of rank r, with Schlafli type {5, 5,6,3r—8,6,6, 6,4}, and with the following CPR graph.
r — 4 r —3 r — 2 r — 1 r — 2
~ ~ " o-O-O
r4
r4
r4
r4
r— 4^ r—3^ r — 2 r — 1 r — 2
0
2
3
3
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. The group Gr-1 is a sesqui-extension of the group given in Theorem 4.8. Hence rr-1 is a string C-group representation. The sggi r0 can be proved to be a string C-group representation using similar techniques to those the proof of the previous theorem. The fact that G0 n Gr-1 = G0,r-1 follows from the fact that Gr-1 is a sesqui-extension of the group given in Theorem 4.8 and the orbits of the respective subgroups.	□
As in the case of odd ranks, from these examples we construct examples of the same rank but for groups of degree n + 4k where k is an integer, by adding a sequence of alternating 0- and 1-edges of length 4k between the 1-edge and the second 2-edge (counting from the left).
Theorem 4.11. If n and r are integers such that n = 3 (mod 4), n > 19 and 8 < r < (n — 1)/2 — 1, r even, then the group An admits a string C-group representation of rank r, with Schlafli type {n — 2(r — 1), 12, 6,3r—8,6,6, 6,4}, and with the following CPR graph.
0q1q0q1q Q 1 Q 2 Q 3 Q C>r—4q t'—3q r — 2^ r— r — 2^
r4
r4
r4
r4
........o^^W^WW5
0
There are two ways to prove this theorem, either by a proof similar to that of Theorem 4.9 or by a proof similar to that of Theorem 4.10. We leave the details to the interested reader.
Theorem 4.12. If n = 3 (mod 4) with n > 15, then the group An admits a string C-group representation of rank 4, with Schlafli type {10, 7,4} for n = 15 and {2(n — 10), 14,4} for n > 15, with the following CPR-graph.
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. The group G0 is isomorphic to 26 : A7 : C2 for n =15 and 26 : A7 : C2 x C2 for n > 19, no matter how big n is. It can easily be checked with Magma that r0 is a string C-group
M. E. Fernandes and D. Leemans: String C-group representations of alternating groups 307
representation for n = 15 and n =19 and since adding more points to the graph will not change the structure of G0, we can conclude that r0 is a string C-group representation for every n > 15. The group G3 acts as Sn-7 on the vertices of the top of the graph and acts as D7 on the remaining vertices, and is a subgroup of (An-7 x D7)+. We can thus conclude that G3 is An-7 x D7. The group G0,3 is isomorphic to D7 for n =15 and C2 x D7 when n > 19 (as there are extra 1-edges in the graph). The group G2,3 is isomorphic to D(n_10). It is obvious from the permutation representation graph that G0 3 n G2 3 is isomorphic to C2. Hence, by Proposition 2.1, the sggi r3 is a string C-group representation. Now, the intersection G0 n G3 = G0,3 need only to be checked in the cases n € {15,19}, which can be done with Magma. Hence, again, by Proposition 2.1, we see that r is a string C-group representation.
It remains to show that G is isomorphic to An. The structure of G3 shows that the action of G3 on the (n - 7) vertices at the top of the graph is An-7. Hence there exists a cycle of order 3 in G0 acting on those vertices. This cycle necessarily fixes the 7 other vertices, so it is a cycle of G. Moreover, that action is (n - 9)-transitive on the top vertices. Hence the stabilizer, in G, of the leftmost vertex of the graph must be transitive on the remaining vertices and G is 2-transitive, therefore primitive. Then, by Theorem 3.4, we can conclude that G > An. Since all generators of G are even permutations, we conclude that G is isomorphic to An.
The Schlafli type follows immediately from the permutation representation graph. □
Theorem 4.13. If n = 3 (mod 4) with n > 15, then the group An admits a string C-group representation of rank 5, with Schlafli type {n —10, 6,6,5}, with the following CPR-graph.
o—o—o—O-O.........o—o—o—o—O^)-o—O-O—o
o—0=4=0^—0—0
2
2
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. The group G0 is isomorphic to S12 no matter how large n is. One can easily check with MAGMA that the permutation representation graph corresponding to r0 is a CPR graph. The group G0,4 is isomorphic to 23 : S3 x S3 no matter how large n is. G3,4 is isomorphic to Sn-9 by Theorem 3.4, as it contains a cycle of length 3, namely (p1 p2)2 and is obviously 2-transitive on n—9 vertices. Moreover, by [10, Theorem 4.1], r3 4 is a string C-group representation as it is generated by three involutions, two of which commute. The group G0 3 4 is isomorphic to D6. Looking at the respective orbits of G0,4 and G3,4 we can conclude that G0,4nG3,4 = G034 and therefore r4 is a string C-group representation. Moreover, one can check that the group G4 is isomorphic to An-8 x C2 : S3 but this is not needed to finish the proof. Now, it is easy to check with MAGMA that G0 n G4 = G0,4 for n =15 and this intersection does not depend on the degree of G. Therefore, by Proposition 2.1, we may conclude that r is a string C-group representation with the given permutation representation graph. A similar argument as in the proof of Theorem 4.12 shows that G is isomorphic to An. The Schlafli type follows immediately from the permutation representation graph.	□
Theorem 4.14. If n = 3 (mod 4) with n > 15, then the group An admits a string C-group representation of rank 6, with Schlafli type {n — 10, 6,3,5,3}, with the following
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CPR-graph.
o—o—o-O-O.........o—o—o—o—o—o—o—O—o
Ô-—-Q—r~0
5	4
2	2 2
3,5
54
Proof. Let r := (G, S) be the sggi having the permutation representation graph above. The group G0 is isomorphic to S12 no matter how big n is. One can easily check with Magma that the permutation representation graph corresponding to r0 is a CPR graph. We have G0,5 = S7 x A5 no matter how big n is. Here G3,4,5 = Sn-9 as proven in the previous theorem (for G34 in the previous theorem is the same group as G3 4 5 here). Similarly, we have G0 4 5 = 22 : S3 x S3. As G3 4 5 n G0 4 5 = G0 3 4 5 independently on how big n is, we can conclude by Proposition 2.1 that r4 5 is a string C-group representation. Similarly, as G0,5 n G4 5 = G0,4,5 no matter how big n is, we can conclude by Proposition 2.1 that r5 is a string C-group representation. Finally, as G0 n G5 = G0,5 no matter how big n is, we conclude that r is a string C-group representation.
It remains to show that G is isomorphic to An. Similar arguments as in the proof of the previous two theorems lead to that conclusion. The Schlafli type follows immediately from the permutation representation graph.	□
Observe that this last family of string C-group representations of rank 6 gives, using the same general construction we used in Theorems 4.2 and 4.7, a family of string C-groups of rank 5 with Schlafli type {n — 10, 6,5,3}.
Theorem 4.15. If n = 3 (mod 4) with n > 15, then the group An admits a string C-group
representation of rank 5, with Schlafli type {n — 9, 6, 5, 3}, with the following CPR-graph. çy^y^y^y^.........
5	4
2	2 2
3,5
54
We leave the proof of this last theorem to the interested reader as it is very similar to the previous proofs.
3
3
4
3
3
4
5 Concluding remarks
Mark Mixer mentioned a similar result in 2015 at the AMS Fall Eastern Sectional Meeting in Rutgers (talk 1115-20-283).
The techniques we developed in this paper inspired Brooksbank and the second author to develop a general rank reduction technique, now available in [4].
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