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4URAL MATHEMATICAL JOURNAL, Vol. 10, No. 1, 2024, pp. 147-158
DOI: 10.15826/umj.2024.1.013
ON G-VERTEX-TRANSITIVE COVERS OF COMPLETE GRAPHS HAVING AT MOST TWO G-ORBITS ON THE ARC SET1
Ludmila Yu. Tsiovkina
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation
Abstract: We investigate abelian (in the sense of Godsil and Hensel) distance-regular covers of complete graphs with the following property: there is a vertex-transitive group of automorphisms of the cover which possesses at most two orbits in the induced action on its arc set. We focus on covers whose parameters belong to some known infinite series of feasible parameters. We also complete the classification of arc-transitive covers with a non-solvable automorphism group and show that the automorphism group of any unknown edge-transitive cover induces a one-dimensional affine permutation group on the set of its antipodal classes.
Keywords: Antipodal cover, Distance-regular graph, Vertex-transitive graph, Arc-transitive graph.
1. Introduction
A distance-regular cover of a complete graph with parameters (n, r, (or an (n, r, -cover) is a connected graph (by "graph" hereafter we mean an undirected graph without loops or multiple edges) whose vertex set can be partitioned into n blocks (or antipodal classes) of equal size r > 2, such that each block induces an r-coclique, the union of any two different blocks induces a perfect matching, and any two non-adjacent vertices in different blocks have exactly ^ > 1 common neighbors. For an (n, r, ^)-cover r, we denote by CG(r) the group of all automorphisms of r that fix each of its antipodal classes setwise. If the group CG(r) is abelian and acts regularly on each antipodal class, then r is called an abelian (n, r, ^)-cover (see [9]).
Abelian covers form an intriguing, large subclass of covers that, by a Godsil-Hensel criterion, can be characterized in terms of certain matrices over group algebras [9]. Only a few general techniques for constructing such covers are known, and typically they require the existence of a related object (e.g., a crooked function, a generalized Hadamard matrix) that is again hard to construct or yield only covers with specific parameters. A promising task is to describe covers with "rich" automorphism groups. There are three known families of abelian covers with arc-transitive automorphism groups, namely, distance-transitive Taylor graphs, Thas-Somma covers, collinearity graphs of certain generalized quadrangles with a deleted spread, and related covers that come from the quotient construction due to Godsil and Hensel. A recent study [12, 14] shows that the list is almost complete, i.e., a new arc-transitive abelian (n, r, ^)-cover may be discovered only in a few open subcases. However, the general problem of classification of abelian (n, r, ^)-covers, whose automorphism group is vertex-transitive and has at most two orbits in its induced action on the arc set of the cover, is far from being resolved. In this paper, we will investigate this problem by focusing on covers with the triple of parameters (n, r, belonging to some infinite series of feasible
1This work was supported by the Russian Science Foundation (project No. 20-71-00122).
parameters. We will also address some open subcases in classification of arc-transitive (n, r, rf)-covers and show that every arc-transitive (n, r, rf)-cover with a non-solvable automorphism group is known and is indeed a member of one of the above-mentioned families whenever it is abelian.
Note that if r is a G-vertex-transitive abelian (n, r, rf)-cover with CG(r) < G such that G induces a rank s permutation group Gs on the set £ of antipodal classes of r, then the arc set of r is partitioned into a collection of s — 1 orbitals of G. Therefore, the arc set of r is the union of at most two orbitals of G if and only if s < 3. This observation allows us to apply the classification of permutation groups of rank at most 3 in our arguments, as well as the classification results on abelian covers with primitive rank 3 groups Gs obtained in [15-17].
The first class of covers considered in this paper (see Section 2) is abelian (n, r, rf)-covers r with parameters of the form
(t — 1)2
(n, r, (i) = ((i2 - l)2, r, (t2 + t- l)^^)
where —t is the smallest eigenvalue of the cover, and r is an odd divisor of t — 1 with gcd(r, 3) = 1. The study of covers with such parameters is motivated by the following result.
Theorem 1 (Coutinho, Godsil, Shirazi, Zhan [6]). Let X be an abelian (n, r, rf)-cover with eigenvalues n — 1 >0> —1 > t , where r is odd and 5 = A — rf. It yields a set of complex equiangular lines for which the absolute bound is attained if and only if either r = ¡j, = 3 = x/n or t = —t € N, gcd(3,r) = 1, r divides t — 1, and the eigenvalues 0,t, and the parameters (n,r, rf, 5) are as follows:
d = (t2 — 2)t, t = —t, n = (t2 — 1)2, rrf = (t — 1)2(t2 +1 — 1) 5 = (t2 — 3)t,
where
me = (t2 — 1) (r — 1), mr = (t2 — 2)(t2 — 1) (r — 1), here ma denotes the multiplicity of the eigenvalue a € {0,t}.
In the case (n,r, rf) = (9,3,3), according to the result of Brouwer and Wilbrink (see [3, p. 386]), there exist exactly two non-isomorphic covers from the conclusion of Theorem 1; they can be constructed by removing one of two non-isomorphic spreads from the (unique) generalized quadrangle GQ(2,4). One of them is distance-transitive, so its automorphism group induces a rank 2 permutation group on the antipodal classes. The automorphism group of the other acts vertex-transitively but intransitively on the arc set, and induces a rank 3 permutation group on the set of antipodal classes. The existence of covers with parameters
(t — 1)2
(n, r, = ((i2 - l)2, r, (t2+t- 1 y-1^)
from the conclusion of this theorem is an open question (see the discussion in [6]). We will describe some basic properties of automorphism groups of covers r with these parameters and apply them to investigate G-vertex-transitive covers r under restrictions on the rank s of the group or on the parameter t: (i) s < 3 or (ii) t < 11.
The second class considered (see Section 3) is abelian covers r, for which the group Gs is an affine group of permutation rank s = 2. A classification of such pairs (r,G) was given in [14]; however, the question of a complete description of pairs (r, G) has remained open in the case when |£| is even and Gs is an affine group of one-dimensional, symplectic or G2 type. We will show that in this case each cover r with r > 2 and a non-solvable group G is known and isomorphic
to a quotient of a certain distance-transitive Thas-Somma cover (see Theorem 2). As a result, we will complete the classification of arc-transitive (n, r, ^)-covers with a non-solvable automorphism group. We will also show that the automorphism group of any unknown edge-transitive (n, r, cover must induce a one-dimensional affine permutation group on the set of its antipodal classes (see Theorem 3).
Our terminology and notation are mostly standard and can be found in [1, 3] and [17].
2. Covers with parameters ((t2 — 1)2, r, (t2 + t — 1)(t — 1)2/r) In this section, r is an abelian cover with parameters
(V - l)2,r, (t2 +t- l)(t - i)^—,
where t is a positive integer, r is an odd divisor of t — 1 such that gcd(3, r) = 1, and with eigenvalues
k = t2(t2 — 2), 9 = (t2 — 2)t, —1, t = —t
of multiplicities
1, m0 = (t2 — 1)(r — 1), k, mr = (t2 — 2)(t2 — 1)(r — 1),
respectively. Let £ be the set of antipodal classes of r, n = |£|, v = nr, a € F € £, and suppose that K = CG(r) < G < Aut(r), G acts transitively on the vertices of r, and put M = G{F} and C = gf . Note that
V = (t3 — 2t + 1)(t — 1) / r, A = (t3 — 2t + 1)(t — 1)/r + t(t2 — 3),
where A denotes the number of common neighbors for two adjacent vertices.
Further, for an element g € G, we will denote by a(g) the number of vertices x of r such that d(x,xg) = i. By n(l) we will denote the set of prime divisors of a positive integer l. For a finite group X, the set n(|X|) is called its prime spectrum and is briefly denoted by n(X). In what follows, if it is clear from the context, for a graph we denote by [x] the adjacency of x in that is, [x] = $1(x).
Lemma 1. The following statements are true:
1) C = CG(K) n Ga and M = K : Ga;
2) |G : M| = (t2 — 1)2 and |G : G0| divides (t2 — 1)2(t — 1);
3) n(G) = n(nr) U n(Go) C ^n! ■ (t — ;
4) |Fix(Go)| = |NG(Go) : Go| divides nr, and |Fixs(M)| = |NG(M) : M| divides n;
5) if p € n(G) and p > t + 1, then p € n(GF);
6) G/Cg(K) < Aut(K) and if r is prime, then G/Cg(K) < Zr-1 and every element g € G of prime order p € n(r — 1) either has no fixed points or Fix(g) is the union of some antipodal classes;
7) if g is an element of prime order p in G, p > r and p € n((t2 — 1)(t2 — 2)t), then r3 = r — 1, r2r1 > 0, and |Q| > r(1 + r1) + r ■ z for some non-negative integer z which is a multiple of p, where r = |r^(a)| (mod p), i = 1,2,3.
Proof straightforwardly follows from the assumptions that r is abelian and G is transitive on its vertices. □
Lemma 2. Let p € n(G). If p > t — 1 and p does not divide t + 1, then
max{p, |Fixs(g)|} < rf;
in particular, if p > rf/2, then for any element g € G of order p, the subgraph Fix(g) is an abelian (|Fix^(g)|, r, rf)-cover with
rf = rf (mod p), rf> |Fixs(g)| > (r — 1)rf', n(r) C n(|Fixs(g)|).
Proof. Let g € G and |g| = p > t — 1. Suppose that p does not divide t + 1. Then Q := Fix(g) = 0 and |Q| = lr, where l = |Fix^(g)|.
(1) Suppose a2(g) > 0. By definition, this means that d(x,xg) = 2 for some vertex x € r — Q. But then [x] n [xg] contains exactly l vertices from Q, implying l < rf.
Assume p > rf. Then [a] n [b] C Q for any non-adjacent vertices a and b from Q belonging to different antipodal classes. But according to [9, Lemma 3.1], Q is a (l, r, rf)-cover and thus l > (r — 1)rf > rf, which is clearly impossible. Therefore, p < rf and since the number rf is composite, p < rf. Notice that if p > rf/2, then
0 < rf := | [a] n [b] n Q| = rf (mod p) < rf/2 < p
for any non-adjacent vertices a and b from Q, belonging to different antipodal classes. Then by [9, Lemma 3.1], Q is an abelian (l, r, rf')-cover, implying rf > l > (r — 1)rf and n(r) C n(l) by [10, Theorem 2.5].
(2) Suppose a2(g) = 0. Then a0(g) = lr = ao(gl) for all 1 < i < p — 1 and ai(g) = r(n — l). If a2(gs) > 0 for some 1 < s < p — 1, then the desired result is obtained by reasoning as in part (1). Suppose that a2(g) = a2(gl) = 0 for all 1 < i < p — 1. Then each (g)-orbit in r — Q is an (n — l)-clique. Since n — l < A + 2, we have l > (r — 1)rf > A. Since |[x] n [xg] n Q| = l for any vertices x and xg from r — Q, we obtain n — l = 2, which means that each (g)-orbit in r — Q is an edge and g2 € Gx, so |g| = 2, a contradiction. The lemma is proven. □
Lemma 3. Suppose g is an involution of G and Q = Fix(g) = 0. Let f = |QnF(z)| for z € Q, l = |Fix^(g)|, and
X = {x € r \ Q| [x] n Q = 0}. Then Q is a regular graph of degree l — 1 on lf vertices and the following statements hold:
1) if l = 1, then t is even, a3(g) = 0, a1(g) + a2(g) = (n — 1)r, and Q coincides with the antipodal class F (z);
2) if l > 1, then 03(g) = (r — f )l, a1(g) + a2(g) = (n — l)r, each vertex in X is "on average" adjacent to fl(n — l)/|X| vertices in Q, and the number of such vertices in Q does not exceed l (in particular, if |X| = f (n — l), then l < A) and
n— l n— l
furthermore, if F(z) C Q, then X = r \ Q and either (i) a1(g) = (n — l)r, l < A, and each (g)-orbit on r \ Q is an edge, or (ii) a2(g) > 0 and l < rf;
3) if t is even, then X = {x € r| d(x,xg) € {1, 2}};
4) if f = 1 and l > 1, then Q is an l-clique and l < rrf/(t — 1) < rf;
5) if f > 1, t is even, and l > 1, then the diameter of the graph Q is 3 and |Q| < l + (l — 1)2(l — 2).
Proof. Let f = |Q n F(z)| and assume Q = 0 (which is automatically satisfied for even t). Then |Q| = f ■ l > 0 and for any two vertices a and b in Q, we have
l — 1 = |[a] n Q| = |[b] n Q|,
meaning that the graph Q is regular of degree l — 1. Recall that
¡i = {t2+t- A = (t2 — l)2 — 2 — (r — l)/x.
Since r is odd and divides t — 1, we have
t — 1 = l = A = ^ (mod 2).
Let A be the set of all vertices from antipodal classes that intersect Q.
Note that if d(x,xg) = 3, then x € A, |F(x) n Q| = f, and [x] C r \ Q. And if x = xg and d(x, xg) < 3, then x € r \ A and |[x] n [xg] n Q| = A = ^ (mod 2). Therefore,
A = Q U {x € r| d(x, xg) = 3}, r \ A = {x € r| d(x, xg) € {1, 2}}.
From the above, we have
«3(g) = (r — f )1, a1(g) + a2(g) = (n — 1)r,
and for each vertex y € r \ A, |[y] n Q| < l, so that either |[y] n Q| < ^ and d(y,yg) = 2, or |[y] n Q| < A and d(y,yg) = 1. On the other hand, each vertex in A is adjacent exactly to n — l vertices in r \ A. Let
X = {x € r \ Q| [x] n Q = 0.
Then
|J [b] n (r \ A) C X,
beF (z)
implying |X| > (n — 1)f, and X = r \ A for even t. Estimating the number of edges between X and Q gives
|Q|(n — l) < min{ 1(«1(g) + «2(g)), A«1 (g) + .
And since
A«1(g) + ^«¡(g) = (A — ^)«1(g) + ^(n — 1)r
and the number of vertices in X does not exceed the number of edges from Q to X, we have |X| < |Q|(n — l) and
< |Q| = f . i < (A~/x)ai(g) + < r max{/i) A} = rA_
n — l n — l
Hence for even t we have |Q| > r, and in particular, for l = 1, we have Q = A.
Let l > 1 and $ be a connected component of the graph Q. Then $ is an f-covering of the complete graph on m = |$|/f vertices, in particular, $ = Q for even t. Since the number of edges from Q to X is |Q|(n — l), each vertex in X is adjacent "on average" to f1(n — 1)/|X| vertices in Q. Recall that ^ < A. Therefore, if |X| = f (n — l), then l < A.
If f = 1, then Q is an l-clique and l < r^/(t — 1). And if f > 1 and t is even, then the diameter of Q is 3 and, due to the Moore bound, we have
|Q| < l + (1 — 1)2(1 — 2).
Suppose f = r. If a1(g) = (n — 1)r, then l < A and each (g)-orbit on r \ A is an edge. If a1(g) < (n — 1)r (which is equivalent to a2(g) > 0), then l < The lemma is proven. □
Lemma 4. Suppose that Gs has permutation rank 3 with subdegrees 1 < k1 < k2. Then
1) Ga has exactly two orbits on [a], denoted X1 and X2, with lengths k1 and k2 respectively, satisfying
k1(A — A1) = k2(A — A2), (2.1)
where Aj = |[xj] n Xj| for vertex xj € Xj, i = 1, 2;
2) if Ga fixes a vertex a* € F(a) — {a}, then
k1(rf — rf1) = k2(rf — rf2), (2.2)
where rf1 = |[y1] n X1| for vertex y1 € X*, rf2 = |[y2] n X*| for vertex y2 € X2, and Xj* is a Ga-orbit on [a*] of length kj, i = 1,2.
P r o o f. As the groups Ga[a] and G{f}S-{f} are permutation isomorphic, (2.1) follows by counting the number of edges between X1 and X2 in two different ways, and (2.2) is obtained in a similar way by counting the number of edges between X* and X2.
□
Proposition 1. The permutation rank s of the group Gs is not equal to 2 (so r cannot be arc-transitive ).
Proof. On the contrary, suppose s = 2. Then, since r is abelian and K < G, we conclude that G acts transitively on the arcs of r. As n is not a prime power, by Burnside's theorem it follows that Gs is an almost simple group. But then the number k = t2(t2 — 2) is a power of a prime (see [12]), a contradiction. □
Proposition 2. If Gs is a primitive group of rank 3, then the group Gs is either almost simple and its socle cannot be an alternating group or a sporadic simple group, or T x T < Gs < T0 I 2, where T0 is a 2-transitive group of degree n0 = t2 — 1 with a simple non-abelian socle T and T cannot be an alternating group of degree n0.
Proof. Suppose that Gs is a primitive group of rank 3 with nontrivial subdegrees k1 and k2, where k1 < k2. Under this condition, according to [5, Theorem 2.6.14] and [2], the group Gs of degree (t2 — 1)2 has two self-paired orbitals on £.
Since n = (t2 — 1)2 is not a prime power, by the classification of primitive groups of rank 3 (e.g., see [4, Chapter 11, Theorem 11.1.1]), the group is either almost simple or is of wreath product type. Furthermore, according to [15, 16], the socle Soc(Gs) of the group Gs cannot be an alternating or a sporadic simple group for all t.
Suppose that the group Gs is of wreath product type, that is, P = T x T < Gs < T0 I 2, where T0 is a 2-transitive group of degree n0 with a simple non-abelian socle T and n = n2. Then nontrivial subdegrees of the group
are k1 = 2(n0 — 1) and k2 = (n0 — 1)2. Simplifying the
equation (2.1), we get
2(A — A1) = (n0 — 1)(A — A2).
Since the group Ga contains a subgroup A = S x S, where S is the stabilizer of a point in T, and A has exactly two orbits on X1, say Y1 and Y2, of lengths |Y1| = |Y21 = n0 — 1, and AYi ~ S, A1 must be a sum of subdegrees of AXl. Hence in the case T ~ Altn0 we have S ~ Altn0-1 and A1 € {0, n0 — 2, n0 — 1,2n0 — 3}. But then the above equation implies that n0 — 1 = t2 — 2 divides
Table 1. Parameters of r with t < 11 (see [6]).
n r ß 6 = A — ß e T = -t me mT
1225 5 205 198 204 -6 140 4760
3969 7 497 488 496 -8 378 23436
14400 5 2620 1298 1309 -11 480 57120
one of 2A or 2(A — 1), which is impossible. So T ^ Altn0 for all t. □
Next we will consider the case t < 11, in which the parameters of r are given in Table 1.
Proposition 3. If t < 11 and Gs is a primitive group of rank s, then s > 3.
Proof. Suppose that Gs is a primitive group of rank 3 with subdegrees 1, fci and k2, where k1 < k2. Let us consider the case t < 11.
Case t = 6. Suppose t = 6 (so n = 1225 = 5272). According to [ , ], it suffices to consider the case where the group is of wreath product type with degree n = nQ. But then no = 35, To is a 2-transitive group of degree 35, and by the classification of finite 2-transitive permutation groups (e.g., see [4, Theorem 11.2.1]), T ~ Altn0, which contradicts to Proposition 2.
Case t = 8. Suppose t = 8 (so n = 3969 = 3472). The case of an almost simple group Gs is impossible, since according to the classification of primitive almost simple groups of rank 3 (e.g., see [4, Chapter 11] or [7, Tables 4-6]), Soc(Gs) cannot be an exceptional simple group or a classical simple group of degree n.
Suppose that the group
is of wreath product type with degree n = nQ. Then n0 = 63 and T0 is a 2-transitive group of degree 63. By Proposition 2, T ^ Altn0, so the classification of 2-transitive groups (e.g., see [4, Theorem 11.2.1]) gives T ~ PSL6(2). But if T ~ PSL6(2), then the subdegrees of AYi on Y are 1, 1 and 60. Therefore,
A1 (mod n0 — 1) € {0,1, n0 — 3, n0 — 2},
and the equation (2.1) has no solution, a contradiction.
Case t = 11. Suppose t = 11 (so n = 14400 = 263252). The case of an almost simple group Gs is impossible, since according to the classification of primitive almost simple groups of rank 3 (e.g., see [4, Chapter 11]), Soc(Gs) cannot be an exceptional simple group or a classical simple group of degree n.
Suppose the group Gs is of wreath product type of degree n = n0. By Proposition 2, T ^ Altn0, so by [4, Theorem 11.2.1] we have T ~ PSp8(2). But if T ~ PSp8(2), then the subdegrees of AYi on Y are 1, 54 and 64. Therefore,
Ai (mod n0 — 1) € {0, 54, 64, n0 — 1, n — 2, 2n0 — 3}.
But A = ^ + 5 = 3918, and the equation (2.1) has no solution, a contradiction.
The proposition is proven. □
Proposition 4. Ift = 6, G := Gs is an imprimitive group of rank s and B is its imprimitivity
_£> _J3
system, then either s > 3 or s = 3 and {|B|, |B|} = {25,49}, where B € B, GB and G are affine 2-transitive groups.
Proof. Suppose that G is a rank 3 imprimitive group with subdegrees 1, k\, and fc2, where k\ < k2. Let B be the unique nont.rivial system of imprimitivity of the group G (see [8, Lemma 3.3]), and fix arbitrarily its block B (of size fci + 1). Then fci + 1 divides
gcd(fc2,(i2-l)2), G < GbB I Sym(£>), _ _
Gb acts 2-transit.ively on B, and G acts 2-transit.ively on B (e.g., see [8, Lemma 3.1]). Let To = Gb and T = Soc(To). Let S be the kernel of the action of G on B, W be the full pre-image of Gb in G and N be the kernel of the action of W on B. Without loss of generality, we assume that a € F € B. Then K < N n S < W and N = K : Na.
Since n = 352, then |B| € {5, 7, 25, 35, 49,175, 245}. If |B| = 35,175 or 245, then applying the classification of finite 2-transitive groups (e.g., see [4, Theorem 11.2.1]) we obtain T ~ Alt(B). But Lemma 2 implies max(n(G)) < 101, so |B| = 35 = |B| and T ~ Soc(GB) ~ Alt(B). In this case Alt34 < W/S < Sym34. On the other hand, W/N < Sym35, in particular, W/N contains an element of order 35. Since the group NS is normal in W, then either S < N or NS/N > Soc(W/N) ~ T. If S < N, then Alt35 < (W/S)/(N/S) ~ W/N, a contradiction. Hence |W : NS| < 2 and the group NS acts 3-transitively on B, so |G{F} : NS{F}| < 2. But then, simplifying the equation (2.1), we get
(A - Ai) = (fci + 1)(A - A2), which contradicts the fact that A1 € {0, k1 — 1} in this case.
If |B| = 5 or 7, then |B| = 175 or 245, and in view of [4, Theorem 11.2.1] we obtain Soc(GB) ~ Alt(B). But max(n(G)) < 101 by Lemma 2, a contradiction. So {|B|, |B|} = {25,49}.
1. Let |B| = 25. Then simplifying the equation (2.1), we get
(A — Ai)=50(A — A2) > 0, A1A2 > 0.
In view of [4, Theorem 11.2.1] we have either T ~ Alt(B) or T ~ E25 and WB is an affine group. But in the first case Ga is 2-transitive on Xi, hence Ai € {0,ki — 1} and the above equation has no solution, a contradiction. Hence T ~ E25. Applying [4, Theorem 11.2.1] again, we obtain that either Soc(GB) ~ Alt(B) or Soc(GB) ~ E4g and GB is an affine group.
Suppose that Soc(GB) ~ Alt(B). Then Alt48 < W/S < Sym48 and W — S contains an element g of order 37, which fixes pointwise 11 blocks in B \ {B}. In view of Lemma 1 g € Gf n N and |Fix^(g)| > 25 ■ 12 = 300, which is a contradiction to Lemma 2.
2. Let |B| = 49. Then ki = 48 and k2 = 49 ■ 24, so the equation (2.1) implies
2(A — Ai) = 49(A — A2) > 0, AiA2 > 0.
In view of the classification of 2-transitive groups (e.g., see [4, Theorem 11.2.1]) we obtain either T ~ Alt(B) or T ~ £49 and WB is an affine group. But in the first case Ga is 2-transitive on Xi, so Ai € {0, ki — 1} and the equation above has no solution, a contradiction. Hence T ~ E49. Applying again [4, Theorem 11.2.1], we obtain either Soc(GB) ~ Alt(B) or Soc(GB) ~ E25 and GB is an affine group.
Suppose Soc(GB) ~ Alt(B). Then Alt24 < W/S < Sym24. On the other hand, w/n < AGL2(7) and G{F}/N < GL2(7). Since |GL2(7)| = 25327 and |B| = 49, N contains an element g of order 13, which fixes pointwise 11 blocks in B \ {B}. In view of Lemma 1 g € Gf n N and |Fix^(g)| > 49 ■ 12 = 588, which is a contradiction to Lemma 2.
The proposition is proven. □
3. Arc-transitive affine covers
Throughout this section, r is an arc-transitive (2e, r, 2e/r)-cover, where r > 2, with eigenvalues k = 2e —1,0, —1, t of multiplicities 1, , k, mT respectively. Let £ be the set of antipodal classes of the graph r, 2e = n = |£| = r^, v = nr, a € F € £, and K = CG(r). Suppose |K| = r, the group G < Aut(r) acts transitively on the arc set of r and induces an affine 2-transitive permutation group Gs on £. Let C = GF, T be the full preimage of the socle of the group Gs in G and L = GaIn [14], it was shown that under these assumptions, the pair (r,G) satisfies one of the following conditions:
1) e = 2dc, d = 3, r divides 2c, T is an elementary abelian group of order 2er, not containing any subgroup of order 2e that is normal in G, and G2(2c) ~ L < Ga;
2) e = 2dc, d > 1, r divides 2c, T is an elementary abelian group of order 2er, not containing any subgroup of order 2e that is normal in G, and Sp2d(2c) ~ L < Ga;
3) Ga < rLi(2e).
In the corresponding cases, we will say that the group Ga is of type G2, of symplectic type, or of one-dimensional type. Moreover, by [14], L < C whenever the group Ga is of symplectic or of G2 type.
Recall that for each subgroup 1 < N < K the quotient graph rN, that is the graph on the set of N-orbits where two orbits are adjacent if and only if there is an edge of r joining them, is a (n,r/|N|,^|N|)-cover [9].
Theorem 2. Suppose r > 2 and the group Ga is of symplectic or of G2 type. Then the (2e, r, 2e/r) -cover r is isomorphic to a quotient of a distance-transitive Thas-Somma (2e, 2c, 2c(Qd-1)) -cover $ for some subgroup U < CG($) of index r. Furthermore, if r = 2c, then r ~ $ and r is characterized by its parameters in the class of arc-transitive distance-regular covers of complete graphs with a non-solvable automorphism group.
Proof. Let L ~ G2(2c) or SpQd(2c), and r = 2s < 2c. First of all, we will determine possible complements to T in TL, then we will find how many classes of conjugate complements to T the group TL can contain, and to which one the group L can belong. From the conditions, it follows that TL centralizes K and (TL/K)' = TL/K. This means that the group TL is a central extension of the group TL/K (e.g., see [1, Ch. 33]).
Let us consider the group T as a GF(2)L-module and K as its L-invariant subspace. Then, with respect to a suitable GF(2)-basis in T (e.g., see [1, 13.2]), the matrices of elements g of the group L, considered as a subgroup of Aut(T) ~ GLe+s(2), have the following form:
( P(0) 0 \
V ^(g) is ) '
where ^ : L ^ GLe(2) is a homomorphism and Is is the identity matrix of order s x s. Since T is irreducible as a GF(2)L-module, we have ^ ^ 0. Let V = T/K and let us fix a nonzero vector w € K. Define the map Yw : L ^ V by the rule gYw = w0(g) + K for all g € L. Then for all g, h € L, we have
(gh)Yw = (gYw )h + hYw,
which means Yw is a cocycle (e.g., se^[1, Ch. 6]). Recall that according to [11], the first cohomology group H1(L,V) of the group L in V is one-dimensional as a GF(2c)-space. Therefore (e.g., see [1, 17.7]), multiplication by a scalar f € GF(2c) gives a set of functions (/Is)^(g), which determines representatives R of all classes of conjugate complements to T in TL.
Case r = 2c. Since [T,L] = T and K < Z(TL), by [1, 17.12] when = 2c, the GF(2)L-module T is the largest among GF(2)L extensions V of the module K by V with the property [ V, L] = V and K < CV (G). Representatives R and R2 of any two classes of complements to T in TL are conjugate in Aut(T) by matrices of the form diag(/e, fIs), where f € GF(2c). This implies that the set of orbital (2e,r, 2e/r) -covers of the group TL with vertex stabilizer R1 gives the set of orbital (2e,r, 2e/r)-covers of the group TL with vertex stabilizer R2, where each graph from the first collection is isomorphic to some graph from the second collection and vice versa. Therefore, it is enough to find the set of all orbital (2e, r, 2e/r) -covers of the group TL with vertex stabilizer L. One of such covers is the distance-transitive (2e,r, 2e/r) -cover $ from the Thas-Somma construction [10, Example 3.6, Proposition 6.2]. According to [10], it is characterized by its parameters in the class of distance-transitive covers. Next we will show that it is also characterized by its parameters in the class of arc-transitive covers with a non-solvable group of automorphisms. Note that G2(2c) < Spe(2c) (e.g., see [18, Theorem 3.7]), and only in the case when d = 3 two different types are permissible for the group Ga.
Let L ~ Sp2d (2c). Since the groups L[a] and are permutation isomorphic, in order to
determine the subdegrees of the group Ga on [a], it suffices to consider the action of the group Sp2d(2c) on its natural module W. Recall that it acts transitively on the set of all hyperbolic pairs in the space W, and for each non-zero vector w in W, there are exactly q2d-1 hyperbolic pairs of the form (w, u). Therefore, the stabilizer of the vector w fixes pointwise q — 1 non-zero vectors from (w) and has q — 1 orbits on W of length q2d-i. If a vertex b € [a] is associated with the vector w, then, since each vertex b* from F(b) — {b} is adjacent to exactly ^ = q2d-i vertices from [a] — [b] and A — ^ = —2, the last statement implies that the group La,b = La,b* acts transitively on each ^-subgraph of the form [a] n [b*].
Suppose r = $. Let Q and S be the self-paired orbitals of the group TL, corresponding to the arc sets of the graphs $ and r, respectively. Then Q(a) = S(a) and r1(a*) = S(a*) = Q(a) = $1(a) for some vertex a* € F(a). Since T is an elementary abelian group, regular on vertices, for some involutions g € T — K and h € K, we have ag € Q(a) and
agh € S(a). Without loss of generality, we may assume that a* = ah, so that (a*)g = agh. Then r1(a) = S(a) = Q(a*) = $1(a*). Since |Q(agh) n Q(a*)| = |S(agh) n S(a)| = A and A < from the action of the group La,a9 = La* ,(a*)9 on £, we obtain Q(agh) n Q(a*) = S(agh) n S(a), which implies Q n S = 0, a contradiction.
Let L ~ G2(2c). Then, to determine the subdegrees of the group Ga on [a], it suffices to consider the action of the group G2(2c) as a subgroup of Spe(2c) on its natural module W (e.g., see [18, 4.3]). Recall that the stabilizer of a 1-dimensional subspace in L has exactly four orbits on the 1-dimensional subspaces of W, and their lengths are 1, q(q + 1), q3(q + 1), and q5. Therefore, the stabilizer of a non-zero vector w in L fixes pointwise q — 1 non-zero vectors from (w) and has q — 1 orbits on W of length q5. By reasoning as above, we obtain the required statement.
Further, according to [9, Theorem 6.2], for any subgroup U in
K of index 21 := |K : U| > 1, the
quotient graph $U is a (2e, 21,2c-1^)-cover, and the group TL/U acts as an arc-transitive group of automorphisms of this graph. Next we will show that these covers exhaust all possibilities for r when s < c.
Case r < 2c. In this case, s < c. Let us show that the only candidates for r are the quotient graphs of the distance-transitive Thas-Somma (2e, 2c, 2c(2d-1))-cover $ defined for subgroups U < CG($) of index 2s. Let ^ = $U be such a quotient graph, and let Q and S be the self-paired orbitals of the group TL, corresponding to the arc sets of the graphs ^ and r, respectively.
Suppose r = A contradiction is obtained by a similar argument as before. Indeed, then Q(a) = S(a) and r1(a*) = S(a*) = Q(a) = ^1(a) for some vertex a* € F(a). Since T is an elementary abelian group, regular on vertices, for some involutions g € T — K and h € K, we have ag € Q(a) and agh € S(a). Without loss of generality, we may assume that a* = ah, so that
(a*)g = agh Then ri(a) = S(a) = Q(a*) = tf i(a*). Since
|Q(agh) n Q(a*)| = |S(agh) n S(a)| = A = ^ — 2 = qQd-i2c-s — 2, and A is not divisible by q2d-i, from the action of the group La,ag = La*,(a*)9 on £, we obtain
(Q(agh) n Q(a*)) n tS(agh) n S(a)) = 0,
which implies Q n S = 0, a contradiction.
The rest statements of theorem follow from the classification of edge-transitive covers in the almost simple case (see [12]) and in the affine case (see [14]). □
Theorem 2 together with the results of [13, 14] and [12] yields the main theorem of this section.
Theorem 3. If $ is an unknown edge-transitive (n, r, -cover, then the group Aut($) induces a one-dimensional affine permutation group on the set of its antipodal classes. Moreover, if the automorphism group of the cover $ acts transitively on its arc set, then it is solvable, ^ > 1, n = r^ is a prime power, and |CG($)| = r.
4. Concluding remarks
This paper finalizes a major part of the project of classifying edge-transitive distance-regular covers of complete graphs, providing a complete classification in the case when the automorphism group of such cover is non-solvable and arc-transitive.
In general, a larger class of G-vertex-transitive covers having at most two G-orbits in the induced action of G on the arc set is still not well understood, and its study remains relevant in the context of finding new constructions of covers with various parameters of interest, e.g., of abelian covers with parameters ((t2 — 1)2, r, (t2 + t — 1)(t — 1)2/r) whose study is motivated by important problems in discrete geometry [6]. In this paper, we established some basic properties of automorphism groups of abelian covers r with these parameters and applied them to investigate G-vertex-transitive covers r under restrictions on the rank s of the group Gs or the parameter t: s < 3or t < 11. This work will be extended in a forthcoming publication of the author for covers with arbitrary values of t.
Next we list a few open questions:
1) Classify arc-transitive covers with a solvable automorphism group.
2) Classify half-transitive covers.
3) Is there any G-vertex-transitive cover with parameters ((t2 — 1)2,r, (t2 + t — 1)(t — 1)2/r) having precisely two G-orbits in the induced action of G on the arc set?
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