ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.)
ARS MATHEMATICA CONTEMPORANEA 21 (2021) #P1.08 / 105–124
https://doi.org/10.26493/1855-3974.2450.1dc
(Also available at http://amc-journal.eu)
On 2-closures of rank 3 groups
Saveliy V. Skresanov *
Sobolev Institute of Mathematics, 4 Acad. Koptyug avenue, 630090, Novosibirsk, Russia
Novosibirsk State University, 1 Pirogova street, 630090 Novosibirsk, Russia
Received 29 September 2020, accepted 2 April 2021, published online 23 August 2021
Abstract
A permutation group G on Ω is called a rank 3 group if it has precisely three orbits in
its induced action on Ω × Ω. The largest permutation group on Ω having the same orbits
as G on Ω × Ω is called the 2-closure of G. A description of 2-closures of rank 3 groups
is given. As a special case, it is proved that the 2-closure of a primitive one-dimensional
affine rank 3 group of sufficiently large degree is also affine and one-dimensional.
Keywords: 2-closure, rank 3 group, rank 3 graph, permutation group.
Math. Subj. Class. (2020): 20B25, 20B15, 05E18
1 Introduction
Let G be a permutation group on a finite set Ω. Recall that the rank of G is the number of
orbits in the induced action ofG on Ω×Ω; these orbits are called 2-orbits. If a rank 3 group
has even order, then its non-diagonal 2-orbit induces a strongly regular graph on Ω, which
is called a rank 3 graph. It is readily seen that a rank 3 group acts on the corresponding
rank 3 graph as an automorphism group. Notice that an arc-transitive strongly regular graph
need not be a rank 3 graph, since its automorphism group might be intransitive on non-arcs.
Related to this is the notion of a 2-closure of a permutation group [31]. The 2-closure
G(2) of a permutation group G is the largest permutation group having the same 2-orbits
as G. Clearly G ≤ G(2), the 2-closure of G(2) is equal to G(2), and G(2) has the same rank
as G. Note also that given a rank 3 graph Γ corresponding to the rank 3 group G, we have
Aut(Γ) = G(2).
The rank 3 groups are completely classified. A primitive rank 3 group either stabilizes
a nontrivial product decomposition, or is almost simple or is an affine group. The rank 3
groups stabilizing a nontrivial product decomposition are given by the classification of the
*The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1613
with the Ministry of Science and Higher Education of the Russian Federation.
E-mail address: skresan@math.nsc.ru (Saveliy V. Skresanov)
cb This work is licensed under https://creativecommons.org/licenses/by/4.0/
106 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
2-transitive almost simple groups, see Theorem 4.1 (ii)(a) and Section 5 in [6]. Almost
simple rank 3 groups were determined in [1] when the socle is an alternating group, in [16]
when the socle is a classical group and in [21] when the socle is an exceptional or sporadic
group. The classification of affine rank 3 groups was completed in [19].
In order to describe the 2-closures of rank 3 groups (or, equivalently, the automorphism
groups of rank 3 graphs), it is essential to know which rank 3 groups give rise to isomorphic
graphs. Despite all the rank 3 groups being known, it is not a trivial task (considerable
progress in describing rank 3 graphs was made in [4]). In the present paper we give a
detailed description of the 2-closures of rank 3 groups.
Theorem 1.1. Let G be a rank 3 permutation group on a set Ω. Then either G is one of the
groups from Table 8, or exactly one of the following is true.
(i) G is imprimitive, i.e. it preserves a nontrivial decomposition Ω = ∆ × X . Then
G(2) = Sym(∆) ≀ Sym(X).
(ii) G is primitive and preserves a product decomposition Ω = ∆2. Then G(2) =
Sym(∆) ↑ Sym(2).
(iii) G is primitive almost simple with socle L, i.e. L ⊴ G ≤ Aut(L). Then G(2) =
NSym(Ω)(L), and G(2) is almost simple with socle L.
(iv) G is a primitive affine group which does not stabilize a product decomposition. Then
G(2) is also an affine group. More precisely, there exist an integer a ≥ 1 and a prime
power q such that G ≤ AΓLa(q), and exactly one of the following holds (setting
F = GF(q)).
(a) G ≤ AΓL1(q). Then G(2) ≤ AΓL1(q).
(b) G ≤ AΓL2m(q) preserves the bilinear forms graph Hq(2,m), m ≥ 3. Then
G(2) = F 2m ⋊ ((GL2(q) ◦GLm(q))⋊Aut(F )).
(c) G ≤ AΓL2m(q) preserves the affine polar graph VOϵ2m(q), m ≥ 2, ϵ = ±.
Then
G(2) = F 2m ⋊ ΓOϵ2m(q).
(d) G ≤ AΓL10(q) preserves the alternating forms graph A(5, q). Then
G(2) = F 10 ⋊ ((ΓL5(q)/{±1})× (F×/(F×)2)).
(e) G ≤ AΓL16(q) preserves the affine half spin graph VD5,5(q). Then G(2) ≤
AΓL16(q) and
G(2) = F 16 ⋊ ((F× ◦ Inndiag(D5(q)))⋊Aut(F )).
(f) G ≤ AΓL4(q) preserves the Suzuki-Tits ovoid graph VSz(q), q = 22e+1,
e ≥ 1. Then
G(2) = F 4 ⋊ ((F× × Sz(q))⋊Aut(F )).
S. V. Skresanov: On 2-closures of rank 3 groups 107
Up arrow symbol in (ii) denotes the primitive wreath product (see Section 2), notation
for graphs in the affine case is explained in Section 3. Table 8 contains finitely many
permutation groups and can be found in Appendix. We note that the largest degree of a
permutation group from Table 8 is 312.
We also remark that the value of a in (iv) of Theorem 1.1 is not necessarily minimal
subject to G ≤ AΓLa(q), since it is not completely defined by the corresponding rank 3
graph and may depend on the group-theoretical structure of G. Minimal values of a can be
found in Table 1.
The proof of Theorem 1.1 can be divided into three parts. First we reduce the study to
the case when G(2) has the same socle as G, and deal with cases (i)–(iii) (Proposition 2.8).
In the affine case (iv) we apply the classification of affine rank 3 groups [19], and compare
subdegrees of groups from various classes (Lemma 3.5 and Proposition 3.6); that allows
us to deal with case (a). Finally, we invoke known results on automorphisms of some
families of strongly regular graphs to cover cases (b)–(d), while cases (e) and (f) are treated
separately.
The case (iv), (a) of Theorem 1.1 can be formulated as a standalone result that may be
of the independent interest.
Theorem 1.2. Let G be a primitive affine permutation group of rank 3 and suppose that
G ≤ AΓL1(q) for some prime power q. Then G(2) ≤ AΓL1(q), unless the degree and the
smallest subdegree of G are as in Table 7.
It is important to stress that the group G in Theorem 1.2 can have a nonsolvable 2-
closure; such an example of degree 26 has been found in [28].
The main motivation for the present study is the application of Theorem 1.1 to the
computational 2-closure problem. Namely, the problem asks if given generators of a rank 3
group one can find generators of its 2-closure in polynomial time. This task influenced the
scope of the present paper considerably, for instance, while one can determine the structure
of the normalizer in Theorem 1.1 (iii) explicitly depending on the type of the corresponding
rank 3 graph, this is not required for the computational problem as this normalizer can be
computed in polynomial time [22]. In other cases it is possible to compute automorphism
groups of associated rank 3 graphs directly (for example, of Hamming graphs), but in many
situations a more detailed study of relevant groups is required. The author plans to turn to
the computational problem in his future work.
Finally, the author would like to express his gratitude to professors M. Grechkoseeva,
I. Ponomarenko, A. Vasil’ev and to the anonymous referees for numerous helpful com-
ments and suggestions.
2 Reduction to affine case
We will prove Theorem 1.1 by dealing with rank 3 groups on a case by case basis. Recall
the following well-known general classification of rank 3 groups.
Proposition 2.1. Let G be a rank 3 group with socle L. Then G is transitive and one of the
following holds:
(i) G is imprimitive,
(ii) L is a direct product of two isomorphic simple groups, and G preserves a nontrivial
product decomposition,
108 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
(iii) L is nonabelian simple,
(iv) L is elementary abelian.
Proof. Transitivity part is clear. If G is primitive, Theorem 4.1 and Proposition 5.1 from
[6] imply that G belongs to one of the last three cases from the statement.
Suppose that G ≤ Sym(Ω). Observe that G acts imprimitively on Ω if and only if it
preserves a nontrivial decomposition Ω = ∆×X , i.e. the action domain Ω can be identified
with a nontrivial Cartesian product ∆×X , |∆| > 1, |X| > 1, where G permutes blocks of
the form ∆ × {x}, x ∈ X . Denote by Sym(∆) ≀ Sym(X) ≤ Sym(Ω) the wreath product
of Sym(∆) and Sym(X) in the imprimitive action, so G ≤ Sym(∆) ≀ Sym(X).
Proposition 2.2. Let G be an imprimitive rank 3 permutation group on Ω. Let ∆ be a
nontrivial block of imprimitivity of G, so Ω can be identified with ∆ ×X for some set X .
Then G(2) = Sym(∆) ≀ Sym(X).
Proof. Set H = Sym(∆) ≀ Sym(X). Then G ≤ H and since G and H are both groups of
rank 3, we have G(2) = H(2). By [15, Lemma 2.5] (see also [8, Proposition 3.1]), we have
(Sym(∆) ≀ Sym(X))(2) = Sym(∆)(2) ≀ Sym(X)(2) = Sym(∆) ≀ Sym(X),
so H is 2-closed. Hence G(2) = H(2) = H , as claimed.
Suppose that the action domain is a Cartesian power of some set: Ω = ∆m, m ≥ 2 and
|∆| > 1. Denote by Sym(∆) ↑ Sym(m) the wreath product of Sym(∆) and Sym(m) in
the product action, i.e. the base group acts on ∆m coordinatewise, while Sym(m) permutes
the coordinates. We say that G ≤ Sym(Ω) preserves a nontrivial product decomposition
Ω = ∆m if G ≤ Sym(∆) ↑ Sym(m).
If G preserves a nontrivial product decomposition Ω = ∆m, then G induces a per-
mutation group G0 ≤ Sym(∆). Recall that we can identify G with a subgroup of G0 ↑
Sym(m). We need the following well-known formula for the rank of a primitive wreath
product; the proof is provided for completeness (see also [18]).
Lemma 2.3. Let G be a transitive group of rank r. Then G ↑ Sym(m) has rank
(
r+m−1
m
)
.
Proof. Let G ≤ Sym(∆), and recall that Γ = G ↑ Sym(m) acts on ∆m. Choose α1 ∈ ∆
and set α1 = (α1, . . . , α1) ∈ ∆m. Let α1, . . . , αr be representatives of orbits of Gα1 on
∆. Since the point stabilizer Γα1 is equal to Gα1 ↑ Sym(m), the points (αi1 , . . . , αim),
1 ≤ i1 ≤ · · · ≤ im ≤ r, form a set of representatives of orbits of Γα1 on ∆m. The
number of indices i1, . . . , im satisfying 1 ≤ i1 ≤ · · · ≤ im ≤ r is equal to the number of
weak compositions of m into r parts, hence the claim is proved.
Observe that in the particular case when Ω = ∆2, the wreath product Sym(∆) ↑
Sym(2) has rank 3 and its 2-orbit of size |∆|2(|∆| − 1) is the edge set of the Hamming
graph H(2, |∆|).
Proposition 2.4. Let G be a primitive rank 3 permutation group on Ω, preserving a non-
trivial product decomposition Ω = ∆m, m ≥ 2. Then m = 2, a 2-orbit of G induces a
Hamming graph and G(2) = Sym(∆) ↑ Sym(2).
S. V. Skresanov: On 2-closures of rank 3 groups 109
Proof. SetH = Sym(∆) ↑ Sym(m), and recall that by Lemma 2.3,H has rank
(
2+m−1
m
)
=
m+ 1 as a permutation group. Since G ≤ H , we have m+ 1 ≤ 3. Therefore m = 2 and
H is a rank 3 group. Then G(2) = H(2) and it suffices to show that H is 2-closed.
A 2-orbit of H induces the Hamming graph H(2, q) on Ω, where q = |∆|. By [3,
Theorem 9.2.1], Aut(H(2, q)) = Sym(q) ↑ Sym(2). It readily follows that H(2) =
Aut(H(2, q)) = H , completing the proof.
In order to find 2-closures in the last two cases of Proposition 2.1, we need to show that
2-closure almost always preserves the socle of a rank 3 group.
Lemma 2.5. LetG be a primitive rank 3 group and suppose thatG andG(2) have different
socles. Then either G preserves a nontrivial product decomposition, or G is an almost
simple group with socle and degree as in Table 8.
Proof. From [25, Theorem 2] it follows that eitherG preserves a nontrivial product decom-
position, or G and G(2) are almost simple groups. By [20, Theorem 1], the latter situation
applies only to a finite number of rank 3 groups, namely, either G is one of exceptional
examples from [20, Table 1], or the socle of G is G2(q), q ≥ 3, or the socle is Ω7(q). Since
rank 3 graphs are distance-transitive, [20, Proposition 1] implies q ∈ {3, 4, 8} in the case
of G2(q), while [20, Proposition 2] yields q ∈ {2, 3} in the case of Ω7(q).
Lemma 2.6. LetG be a primitive rank 3 group with nonabelian simple socle. ThenG does
not preserve a nontrivial product decomposition.
Proof. LetG ≤ Sym(Ω) andL be the socle ofG. Suppose on the contrary thatG preserves
a nontrivial product decomposition. Since G is primitive, and L is a nonabelian simple
minimal normal subgroup of G, [26, Theorem 8.21] implies that either L is A6 and |Ω| =
36, or L = M12 and |Ω| = 144, or L = Sp4(q), q ≥ 4, q even and |Ω| = q4(q2 − 1)2.
One can easily check that neither of these situations occurs in rank 3 by inspecting the
classification of almost simple rank 3 groups. The reader is referred to [5, Table 5] for
alternating socles, [5, Table 9] for sporadic socles and [5, Tables 6 and 7] for classical
socles.
It should be noted that an almost simple group with rank larger than 3 might preserve a
nontrivial product decomposition, see [26, Section 1.3].
Proposition 2.7. Let G be a primitive rank 3 permutation group on Ω with nonabelian
simple socle L. Then either G appears in Table 8, or G(2) has socle L and G(2) =
NSym(Ω)(L).
Proof. By Lemma 2.6, G does not preserve a nontrivial product decomposition, hence by
Lemma 2.5, either G belongs to Table 8, or 2-closure G(2) has the same socle as G. Set
N = NSym(Ω)(L). Clearly G(2) ≤ N , and to establish equality it suffices to show that N
is a rank 3 group.
Suppose that this not the case and N is 2-transitive. By [6, Proposition 5.2], N has
a unique minimal normal subgroup, and since L is a minimal normal subgroup of N , the
socle of N must be equal to L. Hence N is an almost simple 2-transitive group with
socle L.
The possibilities for a socle of a 2-transitive almost simple group are all known and
moreover, such a socle is a 2-transitive group itself, unless G acts on 28 points and L =
110 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
PSL2(8) (see Theorem 5.3 (S) and the following notes in [6]). By [16, Theorem 1.2], there
is no rank 3 group of degree 28 with socle PSL2(8), hence L and thus G are 2-transitive,
which is a contradiction. Therefore N is a rank 3 group and G(2) = N .
We summarize the results of this section in the following.
Proposition 2.8. Let G be a rank 3 permutation group on Ω. Then either G appears in
Table 8, or exactly one of the following holds.
(i) G is imprimitive, i.e. it preserves a nontrivial decomposition Ω = ∆ × X . Then
G(2) = Sym(∆) ≀ Sym(X).
(ii) G is primitive and preserves a product decomposition Ω = ∆2. Then G(2) =
Sym(∆) ↑ Sym(2).
(iii) G is a primitive almost simple group with socle L, i.e. L ⊴ G ≤ Aut(L). Then
G(2) = NSym(Ω)(L), and G(2) is almost simple with socle L.
(iv) G is a primitive affine group which does not stabilize a product decomposition. Then
G(2) is also an affine group.
3 Affine case
In the previous section we reduced the task of describing the 2-closures of rank 3 groups
to the case when the group in question is affine. Recall that a primitive permutation group
G ≤ Sym(Ω) is called affine, if it has a unique minimal normal subgroup V equal to its
socle, such that V is an elementary abelian p-group for some prime p and G = V ⋊ G0
for some G0 < G. The permutation domain Ω can be identified with V in such a way that
V acts on it by translations, and G0 acts on it as a subgroup of GL(V ). Clearly G0 is the
stabilizer of the zero vector in V under such identification.
If G0 acts semilinearly on V as a GF(q)-vector space, where q is a power of p, then we
write G0 ≤ ΓLm(q), where ΓLm(q) is the full semilinear group and V ≃ GF(q)m. If the
field is clear from the context, we may use ΓL(V ) = ΓLm(q) instead. We write AΓLm(q)
for the full affine semilinear group.
Now we are ready to state the classification of affine rank 3 groups.
Theorem 3.1 ([19]). Let G be a finite primitive affine permutation group of rank 3 and
degree n = pd, with socle V ≃ GF(p)d for some prime p, and let G0 be the stabilizer of
the zero vector in V . Then G0 belongs to one of the following classes.
(A) Infinite classes. These are:
(1) G0 ≤ ΓL1(pd);
(2) G0 is imprimitive as a linear group;
(3) G0 stabilizes the decomposition of V ≃ GF(q)2m into V = V1 ⊗ V2, where
pd = q2m, dimV1 = 2 and dimV2 = m;
(4) G0 ⊵ SLm(
√
q) and pd = qm, where 2 divides dm ;
(5) G0 ⊵ SL2( 3
√
q) and pd = q2, where 3 divides d2 ;
(6) G0 ⊵ SUm(q) and pd = q2m;
S. V. Skresanov: On 2-closures of rank 3 groups 111
(7) G0 ⊵ Ω±2m(q) and p
d = q2m;
(8) G0 ⊵ SL5(q) and pd = q10;
(9) G0 ⊵B3(q) and pd = q8;
(10) G0 ⊵D5(q) and pd = q16;
(11) G0 ⊵ Sz(q) and pd = q4.
(B) ‘Extraspecial’ classes.
(C) ‘Exceptional’ classes.
Moreover, classes (B) and (C) consist of finitely many groups.
Observe that the only case when a primitive affine rank 3 group can lie in some other
class from the statement of Proposition 2.8 is when it preserves a nontrivial product de-
composition. This is precisely case (A2) of the classification, and this situation does occur.
Recall that each rank 3 group gives rise to a rank 3 graph. By [4, Table 11.4], the groups
from case (A) of Theorem 3.1 correspond to the following series of graphs:
• One-dimensional affine graphs (i.e. those arising from case (A1)). These graphs are
either Van Lint–Schrijver, Paley or Peisert graphs [23];
• Hamming graphs. These graphs correspond to linearly imprimitive groups;
• Bilinear forms graph Hq(2,m), where m ≥ 2 and q is a prime power. These graphs
correspond to groups fixing a nontrivial tensor decomposition;
• Affine polar graph VOϵ2m(q), where m ≥ 2, ϵ = ±, and q is a prime power;
• Alternating forms graph A(5, q), where q is a prime power;
• Affine half spin graph VD5,5(q), where q is a prime power;
• Suzuki-Tits ovoid graph VSz(q), where q = 22e+1, e ≥ 1.
The reader is referred to [4] for the construction and basic properties of the mentioned
graphs.
It should be noted that different cases of Theorem 3.1 may lead to isomorphic graphs.
Table 3 lists affine rank 3 groups from case (A) and indicates the corresponding rank 3
graphs. In Tables 1 and 2 we provide degrees and subdegrees of affine rank 3 groups in
case (A). These and some other relevant tables and comments on sources of data used are
collected in Appendix.
Our first goal is to show that almost all pairs of affine rank 3 graphs can be distinguished
based on their subdegrees. We start with the class (A1). The following lemma summarizes
some of the arithmetical conditions for the subdegrees of the corresponding groups.
Lemma 3.2. Let G be a primitive affine rank 3 group from class (A1), having degree
n = pd, where p is a prime. Denote by m1,m2 the subdegrees of G and suppose that
m1 < m2. Then m1 divides m2 and m2m1 divides d.
Proof. See [10, Proposition 3.3] for the first claim and [10, Theorem 3.7, (4)] for the sec-
ond.
112 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
The following lemmas apply conditions from Lemma 3.2 to groups from classes (B),
(C) and (A).
Lemma 3.3. Let G be a primitive affine rank 3 group from class (B). Suppose that G
has the same subdegrees as a group from class (A1). Then the degree and subdegrees of
G are one of the following: (72, 24, 24), (172, 96, 192), (232, 264, 264), (36, 104, 624),
(472, 1104, 1104), (34, 16, 64), (74, 480, 1920).
Proof. Let n denote the degree of G, and let m1 ≤ m2 be the subdegrees. In Table 5 all
possible subdegrees of groups from class (B) are listed. We apply Lemma 3.2. For instance,
if n = 292 then m1 = 168, m2 = 672. The quotient m2m1 = 4 does not divide 2, hence this
case cannot happen. The other cases are treated in the same manner.
Lemma 3.4. Let G be a primitive affine rank 3 group from class (C). Suppose that G has
the same subdegrees as a group from class (A1). Then the degree and subdegrees of G are
(34, 40, 40) or (892, 2640, 5280).
Proof. Follows from Lemma 3.2 and Table 6.
Lemma 3.5. LetG be a primitive affine rank 3 group from class (A) and suppose thatG has
the same subdegrees as a group from class (A1). Then either G lies in (A1) or degree and
subdegrees ofG are one of the following: (32, 4, 4), (34, 16, 64), (36, 104, 624), (24, 5, 10),
(26, 21, 42), (28, 51, 204), (210, 93, 930), (212, 315, 3780), (216, 3855, 61680), (52, 8, 16).
Proof. Suppose that G does not lie in class (A1), but shares subdegrees with some group
from (A1). Notice that in cases (A3) through (A11), exactly one of the subdegrees is
divisible by p, so the subdegrees are not equal (see Table 1). In case (A2) subdegrees are
the same if and only if pm = 3, and consequentially n = 9. This situation is the first
example in our list of parameters, hence from now on we may assume that the subdegrees
of G are not equal.
Let m1 and m2 denote the subdegrees of G, where, as shown earlier, we may assume
m1 < m2. Sincem1 andm2 are subdegrees of some group from the class (A1), Lemma 3.2
yields that m1 divides m2 and the number u = m2m1 divides d, where n = p
d.
Now, since G belongs to one of the classes (A2)–(A11), we apply the above arithmeti-
cal conditions in each case. We consider some classes together, since they give rise to
isomorphic rank 3 graphs and hence have the same formulae for subdegrees. The reader is
referred to Table 1 for the list of subdegrees in question.
(A2) Subdegrees in this case are 2(pm−1) and (pm−1)2. If 2(pm−1) > (pm−1)2,
then pm = 2 and n = 4. It can be easily seen that G is not primitive in this
case, contrary to our hypothesis. Therefore we can assume that 2(pm − 1) <
(pm − 1)2.
Then u = p
m−1
2 and since u divides d = 2m, we have p
m − 1 ≤ 4m. It
follows that (n,m1,m2) is one of (32, 4, 4), (34, 16, 64) or (52, 8, 16).
(A3)–(A5) We write r for the highest power of p dividing m2, so the second subdegree is
equal to r(rm − 1)(rm−1 − 1) for some m ≥ 2.
We have u = r r
m−1−1
r+1 and hence u ≥
rm−1−1
2 . Now r
2m = pd ≥ p r
m−1−1
2 .
Using inequalitiesm ≥ 2 and p ≥ 2, we obtain 2r8(m−1) ≥ 2rm−1 . Therefore
S. V. Skresanov: On 2-closures of rank 3 groups 113
rm−1 ≤ 44 and there are finitely many choices for r and m. Checking these
values of r and m against original divisibility conditions we yield the follow-
ing possibilities for (n,m1,m2): (26, 21, 42), (210, 93, 930), (212, 315, 3780),
(36, 104, 624).
(A6), (A7) u = qm−1 q−1qm−1±1 . Numbers q
m−1 and qm−1 ± 1 are coprime, so qm−1 ± 1
divides q − 1. That is possible only when m = 2, so we have u = q. Now
2q ≤ pq ≤ pd = q4, so q ≤ 16. Hence we have the following possibilities for
n, m1, m2 in this case: (24, 5, 10), (28, 51, 204), (216, 3855, 61680).
(A8) u = q3 − q2 q+1q2+1 . Since q
2 and q2 +1 are coprime, q2 +1 must divide q+1.
This can not happen, so this case does not occur.
(A9) u = q3 q−1q3+1 . Since q
3 + 1 does not divide q − 1, this case does not occur.
(A10) u = q5 − q3 q
2+1
q3+1 . Since q
3 + 1 does not divide q2 + 1, this case does not
occur.
(A11) u = q and pd = q4. Hence we obtain the same possible parameters as in cases
(A6), (A7).
In all cases considered we either got a contradiction or got one of the possible excep-
tions recorded in the statement. The claim is proved.
As an immediate corollary we derive that 2-closures of primitive rank 3 subgroups of
AΓL1(q) also lie in AΓL1(q) (Theorem 1.2), apart from a finite number of exceptions.
Proof of Theorem 1.2. Suppose that G and G(2) have different socles. Since G is not al-
most simple, Lemma 2.5 implies that G(2) and thus G must preserve a nontrivial product
decomposition. In that situation G has subdegrees of the form 2(
√
n − 1), (
√
n − 1)2, in
particular, G has subdegrees as a group from class (A2) and hence parameters of G are
listed in Lemma 3.5. We may assume that G does not preserve a nontrivial product decom-
position and so G and G(2) have equal socles. The claim now follows from Theorem 3.1
and Lemmas 3.3–3.5.
Note that Lemmas 3.3–3.5 list degrees and subdegrees of possible exceptions to Theo-
rem 1.2; in Table 7 of Appendix we collect these data in one place.
Now we move on to establish a partial analogue of Lemma 3.5 for classes (A2)–(A11).
First we need to recall some notions related to quadratic and bilinear forms.
Let V be a vector space over a field F . Given a symmetric bilinear form f : V × V →
F , the radical of f is rad(f) = {x ∈ V | f(x, y) = 0 for all y}; we say that f is non-
singular, if rad(f) = 0. If κ : V × V → F is a quadratic form with an associated bilinear
form f , then the radical of κ is rad(κ) = rad(f) ∩ {x ∈ V | κ(x) = 0}. We say that κ is
non-singular, if rad(κ) = 0, and we say that κ is non-degenerate, if rad(f) = 0.
If F has odd characteristic, then rad(κ) = rad(f). If F has even characteristic and
κ is non-singular, then the dimension of rad(f) is at most one, f induces a non-singular
alternating form on V/ rad(f) and, hence, the dimension of V/ rad(f) is even (see [32,
Section 3.4.7]). Therefore if the dimension of V is even, then the notions of non-singular
and non-degenerate quadratic forms coincide regardless of the characteristic.
114 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
Now we can describe the affine polar graph VOϵ2m(q), m ≥ 2. Let V be a 2m-
dimensional vector space over GF(q), and let κ : V → GF(q) be a non-singular quadratic
form of type ϵ. Vertices of the graph VOϵ2m(q) are identified with vectors from V , and two
distinct vertices u, v ∈ V are joined by an edge if κ(u − v) = 0. Up to isomorphism,
VOϵ2m(q) does not depend on the form κ.
Allowing some abuse of terminology, we say that subdegrees of a rank 3 graph are
simply subdegrees of the respective rank 3 group.
Proposition 3.6. If two affine rank 3 graphs have the same subdegrees, then they are iso-
morphic apart from the following exceptions:
• graphs arising from affine groups from Table 8,
• VSz(q) and VO−4 (q) for q = 2
2e+1, e ≥ 1,
• Paley and Peisert graphs.
In particular, graphs Hq(2, 2) and VO+4 (q) are isomorphic.
Proof. Since classes (B) and (C) of Theorem 3.1 and all exceptional parameter sets of
Lemma 3.5 are included in Table 8, we may assume that our graphs come from the case
(A) and their subdegrees are not among the exceptions from Lemma 3.5.
By Lemma 3.5, if one of the graphs in question arises from the case (A1), then the
second graph also comes from (A1). By Table 2, Van Lint-Schrijver graph has unequal
subdegrees, while Paley and Peisert graphs have equal subdegrees, hence in this case graphs
are either isomorphic or it is a Paley graph and a Peisert graph. We may now assume that
our graphs do not come from (A1).
Notice that given n = pd for p prime, the largest subdegree of graphs from classes
(A3)–(A11) is divisible by p. This is not the case in class (A2), unless n = 4 with subde-
grees 2 and 1. The corresponding rank 3 group is imprimitive in that situation, contrary to
our assumptions. Thus we may assume that none of the two graphs comes from (A2).
We compare subdegrees of classes (A3)–(A11) and collect the relevant information in
Table 4. Let us explain the procedure in the case Hq(2,m) vs. VO±2m(q) only, since other
cases are treated similarly.
Consider the graph Hq(2,m). The number of its vertices is equal to n = q2m and the
second subdegree is equal to q(qm − 1)(qm−1 − 1). Recall that n = pd for some prime p,
and the largest power of p dividing the second subdegree is q. In the case of the graph
VOϵ2m(q), we have n = q
2m and the largest power of p dividing the second subdegree
is qm−1. We obtain a system of equations
q2m = q2m, q = qm−1,
which is written in the relevant cell of Table 4. We derive that m = mm−1 , and hence
m = m = 2, q = q. Now, the second subdegree for VOϵ4(q) is q(q − 1)(q2 + (−1)ϵ).
Therefore ϵ = +, which gives us the first example of affine rank 3 graphs with same
subdegrees. Other cases are dealt with in the same way.
Now, Table 4 lists two cases when graphs from different classes have the same subde-
grees, namely, Hq(2, 2), VO+4 (q) and VSz(q), VO
−
4 (q). To finish the proof of the propo-
sition, we show that graphs Hq(2, 2) and VO+4 (q) are in fact isomorphic.
S. V. Skresanov: On 2-closures of rank 3 groups 115
Identify vertices ofHq(2, 2) with 2×2 matrices over GF(q), and recall that two vertices
are connected by an edge if the rank of their difference is 1. A nonzero 2 × 2 matrix has
rank 1 precisely when its determinant is zero:
rk
(
u1 u3
u4 u2
)
= 1 ⇐⇒ u1u2 − u3u4 = 0.
It can be easily seen that u1u2 − u3u4 is a non-degenerate quadratic form on GF(q)4, so
Hq(2, 2) is isomorphic to the affine polar graph VOϵ4(q). By comparing subdegrees we
derive that ϵ = +, and we are done.
It should be noted that VSz(q) and VO−4 (q) in fact have the same parameters as strongly
regular graphs (see [5, Table 24]). In Lemma 3.13 we will see that these graphs are actually
not isomorphic since they have non-isomorphic automorphism groups.
Paley and Peisert graphs are generally not isomorphic (see [24]), but have the same
parameters since they are strongly regular and self-complementary (i.e. isomorphic to their
complements).
Recall that in order to describe 2-closures of rank 3 groups it suffices to find full au-
tomorphism groups of corresponding rank 3 graphs. Hamming graphs were dealt with in
Proposition 2.4, and graphs arising in the case (A1) were covered in Theorem 1.2. We are
left with five cases: bilinear forms graph, affine polar graph, alternating forms graph, affine
half spin graph and the Suzuki-Tits ovoid graph. In most of these cases the full automor-
phism group was described earlier in some form, and we state relevant results here.
For two groups G1 and G2 let G1 ◦ G2 denote their central product. Note that the
central product GL(U) ◦GL(W ) has a natural action on the tensor product U ⊗W .
Proposition 3.7 ([3, Theorem 9.5.1]). Let q be a prime power and m ≥ 2. Set G =
Aut(Hq(2,m)) and F = GF(q). If m > 2, then
G = F 2m ⋊ ((GL2(q) ◦GLm(q))⋊Aut(F )).
If m = 2, then
G = F 4 ⋊ (((GL2(q) ◦GL2(q))⋊Aut(F ))⋊ C2),
where the additional automorphism of order 2 exchanges components of simple tensors.
Let V be a vector space endowed with a quadratic form κ. We say that a nonzero vector
v ∈ V is isotropic if κ(v) = 0.
Lemma 3.8 ([27]). Let V be a vector space over some (possibly finite) field F , and suppose
that dimV ≥ 3. Let κ : V → F be a non-singular quadratic form, possessing an isotropic
vector. If f is a permutation of V with the property that
κ(x− y) = 0 ⇔ κ(xf − yf ) = 0,
then f ∈ AΓL(V ) and f : x 7→ xϕ + v, v ∈ V , where ϕ ∈ ΓL(V ) is a semisimilarity of κ,
i.e. there exist λ ∈ F× and α ∈ Aut(F ) such that κ(xϕ) = λκ(x)α for all x ∈ V .
116 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
Denote by ΓOϵ2m(q) the group of all semisimilarities of a non-degenerate quadratic
form of type ϵ on the vector space of dimension 2m over the finite field of order q.
The reader is referred to [17, Sections 2.7 and 2.8] for the structure and properties of
groups ΓOϵ2m(q).
Proposition 3.9. Let q be a prime power and m ≥ 2. Set F = GF(q). Then
Aut(VOϵ2m(q)) = F
2m ⋊ ΓOϵ2m(q), ϵ = ±.
Proof. Recall that the graph VOϵ2m(q) is defined by a vector space V = F
2m over F and a
non-singular (or, equivalently, non-degenerate) quadratic form κ : V → F . Since m ≥ 2,
we have dimV ≥ 3 and κ possesses an isotropic vector. The claim now follows from
Lemma 3.8.
Proposition 3.10 ([3, Theorem 9.5.3]). Let q be a prime power and set F = GF(q). Then
Aut(A(5, q)) = F 10 ⋊ ((ΓL5(q)/{±1})× (F×/(F×)2)).
Denote byD5(q) an orthogonal group of universal type, in particular, recall the formula
|Z(D5(q))| = gcd(4, q5 − 1) (see [7, Table 5]).
Lemma 3.11. Let q be a prime power, q16 = pd, let F = GF(q), V = F 16 and set
G = Aut(VD5,5(q)). Then G = V ⋊G0, and
F× ◦D5(q) ≤ G0 = NGLd(p)(D5(q)),
where D5(q) acts on the spin module. Moreover, G0/F× is an almost simple group and
G0 ≤ ΓL16(q).
Proof. Set H = V ⋊ (F× ◦ D5(q)). By [19, Lemma 2.9], D5(q) has two orbits on the
set of lines P1(V ), so H is an affine rank 3 group of type (A10). Clearly G = H(2) so by
Lemma 2.5, G is an affine rank 3 group. By Proposition 3.6, G belongs to class (A10) and
the main result of [19] implies that G0 ≤ NGLd(p)(D5(q)). By [19, (1.4)], the generalized
Fitting subgroup of G0/F× is simple, hence this quotient group is almost simple. By
Hering’s theorem [12] (see also [19, Appendix 1]), the normalizer NGLd(p)(D5(q)) cannot
be transitive on the nonzero vectors of V , so G0 = NGLd(p)(D5(q)) as claimed.
Finally, let a be the minimal integer such that G0 ≤ ΓLa(pd/a). By Table 1, a = 16,
so the last inclusion follows.
We write Inndiag(D5(q)) for the overgroup of D5(q) in Aut(D5(q)), containing all
diagonal automorphisms.
Proposition 3.12. Let q be a prime power, and set F = GF(q). Then
Aut(VD5,5(q)) = F
16 ⋊ ((F× ◦ Inndiag(D5(q)))⋊Aut(F )).
Proof. We follow [19, Lemma 2.9]. Take K = E6(q) to be of universal type, so that
|Z(K)| = gcd(3, q − 1). The Dynkin diagram of K is:
α1◦ −α3◦ −α4◦
|
◦α2
−α5◦ −α6◦
S. V. Skresanov: On 2-closures of rank 3 groups 117
Let Σ be the set of roots and let xα(t), hα(t) be Chevalley generators of K. Write Xα =
{xα(t)|t ∈ F}. Let P be a parabolic subgroup of K corresponding to the set of roots
{α2, α3, α4, α5, α6}, and let P = UL be its Levi decomposition. Moreover, L = MH
and we may choose P such that
U = ⟨Xα | α ∈ Σ+, α involves α1⟩,
M = ⟨X±αi | 2 ≤ i ≤ 6⟩,
where M is of universal type and H = ⟨hαi(t)|t ∈ F, 1 ≤ i ≤ 6⟩ is the Cartan subgroup.
In [19, Lemma 2.9] it was shown that M ≃ D5(q), the group U is elementary abelian of
order q16 and in fact, it is a spin module for M . By [11, Theorem 2.6.5 (f)], H induces
diagonal automorphisms on M , and by [30, Section 1, B] it induces the full group of
diagonal automorphisms. Recall that for an element h of H we have xα(t)h = xα(k · t)
for some k ∈ F . In particular, diagonal automorphisms of D5(q) commute with the action
of the field F on U .
Let ϕ be a generator of the field automorphisms group of K, and note that one can
identify that group with Aut(F ); in particular, ϕ acts on F under such an identification.
By [11, Theorem 2.5.1 (c)], generators xα(t) and hα(t) are carried to xα(tϕ) and hα(tϕ)
by ϕ, so field automorphisms normalize U , M and H . Furthermore, ϕ induces the full
group of field automorphisms on M .
Set T = L⋊ ⟨ϕ⟩. We have M ⊴T and T induces all field and diagonal automorphisms
on M . Set T = T/Z(K) and M = MZ(K)/Z(K). By [11, Theorem 2.6.5 (e)], the
centralizer CAut(K)(U) is the image of Z(U) in Aut(K). Therefore T acts faithfully on
U , and since |Z(M)| is coprime to |Z(K)|, we derive that M ≃ M ≃ D5(q). Hence we
have an embedding T ≤ GLd(p), where |U | = pd, and, with some abuse of notation, T ≤
NGLd(p)(D5(q)). By Lemma 3.11, the latter normalizer is an almost simple group (modulo
scalars), and thus we have shown that it contains all field and diagonal automorphisms of
D5(q). It is left to show that it does not contain graph automorphisms.
Suppose that a graph automorphism ψ lies in G0 = NGLd(p)(D5(q)), and recall that
M ≃ D5(q). By [19, Lemma 2.9], there is an orbit ∆ of G0 on the nonzero vectors of
U , such that the point stabilizer Mδ , δ ∈ ∆ is a parabolic subgroup of type A4. Since ψ
preserves the orbit ∆ and normalizes M , it must take a point stabilizer Mδ to the point
stabilizer Mδ′ for some δ′ ∈ ∆, in particular, it takes M δ to a conjugate subgroup. That
is impossible, since by [11, Theorem 2.6.5 (c)], automorphism ψ interchanges conjugacy
classes of parabolic subgroups of type A4, so the final claim is proved.
Recall the construction of the graph VSz(q), q = 22e+1, e ≥ 1. Set F = GF(q),
V = F 4 and let σ be an automorphism of F acting as σ(x) = x2
e+1
. Define the subset O
of the projective space P1(V ) by
O = {(0, 0, 1, 0)} ∩ {(x, y, z, 1) | z = xy + x2xσ + yσ},
where vectors are written projectively. The vertex set of VSz(q) is V and two vectors are
connected by an edge, if a line connecting them has a direction in O.
Recall that Sz(q) ≤ GL4(q) is faithfully represented on P1(V ) and induces the group
of all collineations which preserve the Suzuki-Tits ovoid O (see [14, Chapter XI, Theo-
rem 3.3]). Clearly scalar transformations preserve the preimage of O in V , and it can be
118 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
easily seen that Oα = O for any α ∈ Aut(F ). Hence the following group
H = V ⋊ ((F× × Sz(q))⋊Aut(F ))
acts as a group of automorphisms of VSz(q). By [13, Lemma 16.4.6], Sz(q) acts transi-
tively on P1(V ) \O, hence H is a rank 3 group.
We will show that H is the full automorphism group of VSz(q), but first we need to
note the following basic fact.
Lemma 3.13. If q = 22e+1, e ≥ 1, then there is no subgroup of Aut(VO−4 (q)) isomorphic
to Sz(q). In particular, graphs VO−4 (q) and VSz(q) are not isomorphic.
Proof. Suppose the contrary, so that Sz(q) is a subgroup of Aut(VO−4 (q)). By Proposi-
tion 3.9, we have Aut(VO−4 (q)) ≃ V ⋊ ΓO
−
4 (q) for some elementary abelian group V .
Recall that the orthogonal group Ω−4 (q) is a normal subgroup of ΓO
−
4 (q), and the quotient
ΓO−4 (q)/Ω
−
4 (q) is solvable. Clearly V is also solvable, and since Sz(q) is simple, we ob-
tain an embedding of Sz(q) into Ω−4 (q). Yet that is impossible, as can be easily seen by
inspection of maximal subgroups of Ω−4 (q), see, for instance, [2, Table 8.17]. That is a
contradiction, so the first claim is proved.
The second claim follows from the fact that Sz(q) lies in Aut(VSz(q)).
Proposition 3.14. Let q = 22e+1, where e ≥ 1, and set F = GF(q). Then
Aut(VSz(q)) = F 4 ⋊ ((F× × Sz(q))⋊Aut(F )).
Proof. Let H = F 4 ⋊ ((F× × Sz(q))⋊ Aut(F )) be a rank 3 group acting on VSz(q) by
automorphisms. Set G = Aut(VSz(q)) and recall that G = H(2). By Lemma 2.5, G is an
affine group with the same socle asH , and by Proposition 3.6 and Table 3, it follows thatG
lies in class (A7) or (A11), or it is one of the groups from Table 8. It can be easily checked
that there is no group with degree q4 and subdegrees (q2 + 1)(q − 1), q(q2 + 1)(q − 1)
in Table 8, so the last possibility does not happen. By Lemma 3.13, G does not lie in
(A7), so it is a group from class (A11). Denote by H0 and G0 zero stabilizers in H and G
respectively. Notice that H0 ≤ G0.
By Theorem 3.1 and Table 1, we have G0 ≤ ΓL4(q) and Sz(q) ⊴ G0. By [19, (1.4)],
given Z = Z(GL4(q)) ≃ F×, the generalized Fitting subgroup ofG0/(G0∩Z) is a simple
group. Hence G0/(G0 ∩ Z) is an almost simple group with socle Sz(q).
The outer automorphisms group of Sz(q) consists of field automorphisms only (see [7,
Table 5]), so
|G0| ≤ |Z| · |Aut(Sz(q))| ≤ |F×||Sz(q)||Aut(F )|.
Since H0 ≃ (F× × Sz(q)) ⋊ Aut(F ), the order of H0 coincides with the value on right-
hand side of the inequality. Now H0 = G0 and the claim is proved.
Proof of Theorem 1.1. Let G be a rank 3 group, and suppose that G is not listed in
Table 8. By Proposition 2.8, we may assume that G is a primitive affine group which
does not stabilize a product decomposition and, moreover, G(2) is also an affine group.
By Theorem 3.1, G is either a one-dimensional affine group (class (A1)), or preserves a
bilinear forms graph Hq(2,m), m ≥ 2, an affine polar graph VOϵ2m(q), ϵ = ±, m ≥ 2,
alternating forms graph A(5, q), affine half-spin graph VD5,5(q), Suzuki-Tits ovoid graph
VSz(q) or lies in class (B) or (C).
S. V. Skresanov: On 2-closures of rank 3 groups 119
The full automorphism groups of these graphs (i.e. 2-closures of respective groups) are
described in Theorem 1.2 (one-dimensional affine groups), Proposition 3.7 (bilinear forms
graph), Proposition 3.9 (affine polar graph), Proposition 3.10 (alternating forms graph),
Proposition 3.12 (affine half-spin graph) and Proposition 3.14 (Suzuki-Tits ovoid graph).
Notice that we do not need to consider classes (B) and (C) as they are included in Table 8.
Since by Proposition 3.6, the graph Hq(2, 2) is isomorphic to VO+4 (q), we may ex-
clude it from the bilinear forms case. Now it is easy to see that cases considered in Theo-
rem 1.1 (iv) are mutually exclusive. Indeed, it suffices to prove that graphs from different
cases are not isomorphic. By Proposition 3.6, if two affine rank 3 graphs have the same
subdegrees, then they belong to the same case except for VSz(q) and VO−4 (q), q = 2
2e+1,
e ≥ 1 (note that we group one-dimensional affine graphs into one case). By Lemma 3.13,
graphs VSz(q) and VO−4 (q) are not isomorphic, which proves the claim.
Finally, inclusions of the form G ≤ AΓLa(q) can be read off Table 1. Notice that in
some cases we do not give the minimal value of a, for example, if SUm(q) ≤ G lies in class
(A6), then G ≤ AΓLm(q2), but we list the inclusion G ≤ AΓL2m(q). This completes the
proof of the theorem.
ORCID iDs
Saveliy V. Skresanov https://orcid.org/0000-0002-8397-5609
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S. V. Skresanov: On 2-closures of rank 3 groups 121
A Appendix
In this section we collect some relevant tabular data. Table 1 lists information on affine
rank 3 groups from class (A), namely, for each groupG it provides rough group-theoretical
structure (column “Type of G”), degree n and subdegrees. Column “a” gives the smallest
integer a such that the stabilizer of the zero vector G0 lies in ΓLa(pd/a). Most of the
information in Table 1 is taken from [19, Table 12], see also [5, Table 10] for the values
of a.
Table 2 lists the subdegrees of one-dimensional affine rank 3 groups. The first col-
umn specifies the type of graph associated to the group in question, next two columns
provide degree and subdegrees, and the last column lists additional constraints on param-
eters involved. By [23], these graphs turn out to be either Van Lint–Schrijver, Paley or
Peisert graphs. See [29, Section 2] for the parameters of the Van Lint–Schrijver graph;
parameters of Paley and Peisert graphs are computed using the fact that these graphs are
self-complementary.
Table 3 lists rank 3 graphs corresponding to rank 3 groups from classes (A1)–(A11),
cf. [4, Table 11.4]. Terminology and graph notation is mostly consistent with [4], see also
[5, Table 10].
Table 4 records information on when some families of affine rank 3 graphs can have
identical subdegrees, the procedure for building this table being described in Proposi-
tion 3.6. Trivial cases (when two graphs are the same) are not listed, also graphs from
cases (A1) and (A2) are omitted, since they are dealt with separately.
Tables 5 and 6 list degrees and subdegrees of affine rank 3 groups from classes (B)
and (C), without repetitions (i.e. parameter sets are listed only once, regardless of whether
several groups possess same parameters). If the smaller subdegree divides the largest, the
last column gives the respective quotient; otherwise a dash is placed. Information in Ta-
ble 5 is taken from [9, Theorem 1.1] and [19, Table 13], see also [5, Table 11]. Information
in Table 6 before the horizontal line is taken from [10, Theorem 5.3], but notice that we ex-
clude the case of 1192, since 119 is not a prime number (that error was observed by Liebeck
in [19]). Information in Table 6 after the horizontal line is taken from [19, Table 14], with
the correction for the case of Alt(9), where subdegrees should be 120, 135 instead of 105,
150, as noted in [5, Table 12].
Table 7 lists parameters of possible exceptions to Theorem 1.2. The table consists of
three subtables, corresponding to classes (A), (B) and (C) of Theorem 3.1, i.e. values for
the first subtable are taken from Lemma 3.5, for the second from Lemma 3.3, and for the
third from Lemma 3.4. Each subtable lists degrees and smallest subdegrees of possible
exceptions. Notice that parameters of one-dimensional affine rank 3 groups stabilizing a
nontrivial product decomposition are collected in the subtable for the class (A).
Table 8 lists possible exceptions to Theorem 1.1. The first column references the state-
ment where a possible exception first appears, the second column describes the structure of
the group, and the third column gives its degree, either explicitly or by referencing another
table. Notice that we include classes (B) and (C) of Theorem 3.1 in Table 8; corresponding
groups can be found in [19, Table 1 and 2].
Finally, we mention that Tables 7 and 8 list potential exceptions to Theorems 1.2 and 1.1
respectively, in particular, it might be possible to remove some parameter sets and groups
by a more careful analysis.
122 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
Table 1: Class (A) in the classification of affine rank 3 groups
Type of G n = pd a Subdegrees
(A1): G0 < ΓL1(pd) pd 1 See Table 2
(A2): G0 imprimitive p2m 2m 2(pm − 1), (pm − 1)2
(A3): tensor product q2m 2m (q + 1)(qm − 1), q(qm − 1)(qm−1 − 1)
(A4): G0 ⊵ SLm(
√
q) qm m (
√
q + 1)(
√
qm − 1), √q(√qm − 1)(√qm−1 − 1)
(A5): G0 ⊵ SL2( 3
√
q) q2 2 ( 3
√
q + 1)(q − 1), 3√q(q − 1)( 3√q2 − 1)
(A6): G0 ⊵ SUm(q) q2m m
{
(qm − 1)(qm−1 + 1), qm−1(q − 1)(qm − 1), m even
(qm + 1)(qm−1 − 1), qm−1(q − 1)(qm + 1), m odd
(A7): G0 ⊵ Ωϵ2m(q) q
2m 2m
{
(qm − 1)(qm−1 + 1), qm−1(q − 1)(qm − 1), ϵ = +
(qm + 1)(qm−1 − 1), qm−1(q − 1)(qm + 1), ϵ = −
(A8): G0 ⊵ SL5(q) q10 10 (q5 − 1)(q2 + 1), q2(q5 − 1)(q3 − 1)
(A9): G0 ⊵ B3(q) q8 8 (q4 − 1)(q3 + 1), q3(q4 − 1)(q − 1)
(A10): G0 ⊵ D5(q) q16 16 (q8 − 1)(q3 + 1), q3(q8 − 1)(q5 − 1)
(A11): G0 ⊵ Sz(q) q4 4 (q2 + 1)(q − 1), q(q2 + 1)(q − 1)
Table 2: Subdegrees of one-dimensional affine rank 3 groups
Graph Degree Subdegrees Comments
Van Lint–Schrijver q = p(e−1)t 1e (q − 1),
1
e (e − 1)(q − 1) e > 2 is prime, p is primitive (mod e)
Paley q 12 (q − 1),
1
2 (q − 1) q ≡ 1 (mod 4)
Peisert q = p2t 12 (q − 1),
1
2 (q − 1) p ≡ 3 (mod 4)
Table 3: Rank 3 graphs in class (A)
Type of G Graph Comments
(A1): G0 < ΓL1(pd) Van Lint–Schrijver, Paley or Peisert graph
(A2): G0 imprimitive Hamming graph
(A3): tensor product bilinear forms graph Hq(2,m)
(A4): G0 ⊵ SLm(
√
q) bilinear forms graph H√q(2,m) SLm(
√
q) stabilizes an m-dimensional
subspace over GF(
√
q)
(A5): G0 ⊵ SL2( 3
√
q) bilinear forms graph H 3√q(2, 3) SL2( 3
√
q) stabilizes a 2-dimensional
subspace over GF( 3
√
q)
(A6): G0 ⊵ SUm(q) affine polar graph VOϵ2m(q), ϵ = (−1)
m
(A7): G0 ⊵ Ωϵ2m(q) affine polar graph VO
ϵ
2m(q)
(A8): G0 ⊵ SL5(q) alternating forms graph A(5, q)
(A9): G0 ⊵ B3(q) affine polar graph VO
+
8 (q)
(A10): G0 ⊵ D5(q) affine half spin graph VD5,5(q)
(A11): G0 ⊵ Sz(q) Suzuki-Tits ovoid graph VSz(q)
Table 4: Intersections between classes based on subdegrees
VO±2m(q) A(5, q) VD5,5(q) VSz(q)
Hq(2,m)
q2m = q2m
q = qm−1
m = mm−1
m = m = 2, q = q
q2m = q10
q = q2
m = 104
Impossible
q2m = q16
q = q3
m = 83
Impossible
q2m = q4
q = q
m = 2
q(q2 − 1)(q − 1) = q(q2 + 1)(q − 1)
Impossible
VO±2m(q)
q2m = q10
qm−1 = q2
m = 53
Impossible
q2m = q16
qm−1 = q3
m = 85
Impossible
q2m = q4
qm−1 = q
m = 2, q = q
A(5, q)
q10 = q16
q2 = q3
Impossible
q10 = q4
q2 = q
Impossible
VD5,5(q)
q16 = q4
q3 = q
Impossible
S. V. Skresanov: On 2-closures of rank 3 groups 123
Table 5: Subdegrees of rank 3 groups in class (B)
n = pd Subdegrees m1, m2 m2m1 if it is an integer
26 27, 36 —
34 32, 48 —
72 24, 24 1
132 72, 96 —
172 96, 192 2
192 144, 216 —
232 264, 264 1
36 104, 624 6
292 168, 672 4
312 240, 720 3
472 1104, 1104 1
34 16, 64 4
54 240, 384 —
74 480, 1920 4
38 1440, 5120 —
Table 6: Subdegrees of rank 3 groups in class (C)
n = pd Subdegrees m1, m2 m2m1 if it is an integer
34 40, 40 1
312 (31− 1) · 12, (31− 1) · 20 —
412 (41− 1) · 12, (41− 1) · 30 —
74 (72 − 1) · 20, (72 − 1) · 30 —
712 (71− 1) · 12, (71− 1) · 60 5
792 (79− 1) · 20, (79− 1) · 60 3
892 (89− 1) · 30, (89− 1) · 60 2
26 18, 45 —
54 144, 480 —
28 45, 210 —
74 720, 1680 —
28 120, 135 —
28 102, 153 —
36 224, 504 —
74 240, 2160 9
35 22, 220 10
35 110, 132 —
211 276, 1771 —
211 759, 1288 —
312 65520, 465920 —
212 1575, 2520 —
56 7560, 8064 —
124 Ars Math. Contemp. 21 (2021) #P1.08 / 105–124
Table 7: Possible exceptions to Theorem 1.2
(A)
Degree 24 26 28 210 212 216 32 34 36 52
Subdegree 5 21 51 93 315 3855 4 16 104 8
(B)
Degree 34 36 72 74 172 232 472
Subdegree 16 104 24 480 96 264 1104
(C)
Degree 34 892
Subdegree 40 2640
Table 8: Possible exceptions to Theorem 1.1
Appearance Type of group Degree
Lemma 2.5 PΓL2(8) 36
M11 55
M12 66
M23 253
M24 276
Alt(9) 120
G2(q)⊴G q3(q3 − 1)/2, where q ∈ {3, 4, 8}
Ω7(q)⊴G q3(q4 − 1)/ gcd(2, q − 1), where q ∈ {2, 3}
Theorem 3.1 (B) and (C) Tables 5 and 6
Theorem 1.2 G ≤ AΓL1(q) Table 7