Testing isomorphism of central Cayley graphs over almost simple groups in polynomial time
Ilia Ponomarenko, Andrey Vasil'ev

TL;DR
This paper proves that isomorphisms between central Cayley graphs over almost simple groups can be computed efficiently in polynomial time, advancing understanding of graph isomorphism problems in algebraic structures.
Contribution
It establishes a polynomial-time algorithm for testing isomorphism of central Cayley graphs over explicitly given almost simple groups.
Findings
Isomorphism testing is polynomial-time for these graphs.
Algorithm explicitly given for almost simple groups.
Advances the graph isomorphism complexity theory.
Abstract
A Cayley graph over a group G is said to be central if its connection set is a normal subset of G. It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order n, the set of all isomorphisms from the first graph onto the second can be found in time poly(n).
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Testing isomorphism of central Cayley graphs over almost simple groups in polynomial time
Ilia Ponomarenko
St.Petersburg Department of the Steklov Mathematical Institute, St.Petersburg, Russia
and
Andrey Vasil’ev
Sobolev Institute of Mathematics, Novosibirsk, Russia
Novosibirsk State University, Novosibirsk, Russia
In memory of Sergei Evdokimov
Abstract.
A Cayley graph over a group is said to be central if its connection set is a normal subset of . It is proved that for any two central Cayley graphs over explicitly given almost simple groups of order , the set of all isomorphisms from the first graph onto the second can be found in time .
1. Introduction
In the present paper, we are interested in a special case of the following restriction of the Graph Isomorphism Problem to the class of Cayley graphs.
Cayley Graph Isomorphism Problem. For two explicitly given finite groups and and two sets and , construct the set , where and .
Here the input consists of the multiplication tables of and and the sets and , whereas the output is either empty or given by an element of the set and a generating set of the group (of size polynomial in the order of the group ). Obviously, the Luks algorithm [12] solves the Cayley Graph Isomorphism Problem in polynomial time for every group , whenever the set is of constant sizes. If is cyclic, then the problem with no restriction for is also solvable in polynomial time (see [6] and [13]). It should be noted that if and is a so-called CI-group, then an obvious algorithm solves the Cayley Graph Isomorphism Problem in time polynomial in (for details, see [10]).
The aforementioned special case is formed by the two following conditions imposed on the input graphs and groups. First, we assume that is a central Cayley graph over , which means that is a normal subset of , i.e., for every . Second, the group is assumed to be almost simple, i.e., the socle of is a nonabelian simple group. The same conditions are imposed on the graph and group . Even in this rather restrictive case, the problem is still nontrivial; at least the number of possible input graphs is exponential in .
Example. Let be a symmetric group of degree . Then the number of central colored Cayley graphs over is equal to , where is the number of all partitions of the set . Since is approximately equal to , the number is exponential in .
By technical reasons, it is more convenient to deal with colored Cayley graphs. Such a graph is given by a partition of the group into classes with , and can be thought as arc-colored complete graph with vertex set and the th color class of arcs coinciding with the arc set of the Cayley graph , . We say that is the Cayley partition of this graph and denote the latter by . In what follows, all Cayley graphs are assumed to be colored: the graph is treated as for and .
Theorem 1.1**.**
For any two central Cayley graphs and over explicitly given almost simple groups and of order , the set can be found in time .
Corollary 1.2**.**
The automorphism group of a central Cayley graph over an explicitly given almost simple group of order can be found in time .
The proof of Theorem 1.1 is a mix of combinatorial and permutation group techniques. Section 2 provides a relevant background for the combinatorial part including coherent configurations and Cayley schemes. In Section 3, we use a classification of regular almost simple subgroups of primitive groups [11] to prove (Lemma 3.2) that except for one special case, if is a -closed primitive group containing regular almost simple subgroup, then
[TABLE]
where is the subgroup of generated by the holomorph of and the permutation , . We extend this result to non-primitive groups in Sections 4 and 5 by showing that in this case, either formula (1) holds or is a nontrivial generalized wreath product (Theorems 4.1 and 5.1). We apply this fact in Section 6 to the automorphism group of a central Cayley graph over the group to establish that
[TABLE]
where is the socle of and is the restriction to of the setwise stabilizer of in . Note that if is a symmetric group of degree at least , then the group is isomorphic to the group . Thus as a byproduct of (2), we obtain the following generalization of the result [9, Theorem 1.1] on the automorphism group of the Cayley graph , where is a symmetric group and is the set of its transpositions.
Theorem 1.3**.**
Let be a symmetric group of degree at least , a proper normal subset of , and . Then .
In Sections 7 and 8, we develop algorithmic tools to find the above structure of the group with the help of the related Cayley scheme and the group . The main algorithm providing the proof of Theorem 1.1 is given in Section 9.
Notation.
The diagonal of the Cartesian product is denoted by .
For , set .
For and , set .
For and , set .
For a partition of a set and , set to be the relation consisting of the pairs such that meets .
For a set of binary relations, put to be the set of all unions of relations from .
The symmetric and alternating groups on are denoted by and , respectively.
For and , set .
For a group and its subgroup , set and to be the subgroups of induced by left and right multiplications of , respectively, and .
For a group , set to be the subgroup of generated by the group and the permutation .
For a group and a set , the restriction to of the setwise stabilizer of in is denoted by .
For an imprimitivity system of a transitive group , set and to be, respectively, the intersection of all with and the permutation group induced by the action of on .
For a group and a permutation group , set to be the set of all regular subgroups of that are isomorphic to .
2. Coherent configurations and Cayley schemes
This section contains well-known basic facts on coherent configurations. All of them can be found in [7] and papers cited there.
2.1. Main definitions.
Let be a finite set and a partition of . The pair is called a coherent configuration on if the following conditions hold:
- (C1)
, 2. (C2)
, 3. (C3)
given , the number does not depend on the choice of the pair .
The elements of and , and the numbers are called the points and basis relations, and the intersection numbers of , respectively. The numbers and are called the degree and the rank of . The coherent configuration is said to be homogeneous if .
Denote by the set of such that . The elements of are called the fibers of . In view of condition (C1), the set is the disjoint union of all of them. Moreover, for each , there exist uniquely determined fibers and such that . Note that the coherent configuration is homogeneous if and only if .
Let be an equivalence relation and the set of its classes. Given denote by the set of all nonempty relations with . Then the pair
[TABLE]
is a coherent configuration called the restriction of to . This enables to define the restriction of to a set : the corresponding equivalence relation is equal to the union of , where runs over the fibers contained in . Another coherent configuration associated with is obtained as follows. Denote by the set of all nonempty relations , . Then
[TABLE]
is a coherent configuration called the quotient of modulo .
2.2. Combinatorial and algebraic isomorphisms.
A bijection is called the (combinatorial) isomorphism from onto a coherent configuration if the set contains the relation for each . The set of all isomorphisms is denoted by . The group of all isomorphisms of to itself contains a normal subgroup
[TABLE]
called the automorphism group of . Conversely, let be a permutation group, and let be the set of orbits of the component-wise action of on . Then the pair is a coherent configuration; we say that is associated with and denote it by .
According to Wielandt [17], a permutation group on is said to be -closed if it is equal to its -closure
[TABLE]
or, equivalently, if is an automorphism group of a family of binary relations on (such a family can always be chosen as the set of basis relations of a coherent configuration on ). If is -closed and is an imprimitivity system of , then the group is -closed. However, the group is not always -closed.
A bijection is called an algebraic isomorphism from onto if
[TABLE]
In this case, and are said to be algebraically isomorphic. Each isomorphism from onto induces an algebraic isomorphism between these configurations. The set of all isomorphisms inducing the algebraic isomorphism is denoted by . In particular,
[TABLE]
where is the identity mapping on .
An algebraic isomorphism induces a bijection from onto : the union of basis relations of is taken to . This bijection is also denoted by . It preserves the equivalence relations ; moreover, the equivalence relations and have the same number of classes as well as the same multiset of their sizes. In this case, if and are the sets of classes of and , respectively, and , then induces the algebraic isomorphisms
[TABLE]
for a suitable .
2.3. Direct sum and wreath product.
Let and be coherent configurations. Denote by the disjoint union of and , and by the union of the set and the set of all relations and with and . Then the pair
[TABLE]
is a coherent configuration called the direct sum of and . The automorphism group of this configuration equals the direct product acting on the set . Furthermore, if is an algebraic isomorphism from to another coherent configuration, then the latter is also the direct sum and induces algebraic isomorphisms and coinciding with the restrictions of on and , respectively.
Let be a homogeneous coherent configuration, an equivalence relation, and the set of classes of . We say that is the wreath product with respect to if for each such that ,
[TABLE]
In what follows, we always assume that the classes of can be identified with the help of a family of the isomorphisms , , such that
[TABLE]
In this case, is isomorphic to the usual wreath product for all (see [15, p.45]). The automorphism group of this coherent configuration is permutation isomorphic to the wreath product in imprimitive action.
Furthermore, if is an algebraic isomorphism from the wreath product with respect to to another coherent configuration, then the latter is also the wreath product with respect to and induces algebraic isomorphisms and coinciding with the restrictions of on and , respectively, where is the set of classes of and .
2.4. Cayley schemes.
A coherent configuration is called the Cayley scheme over a group if
[TABLE]
In this case, is homogeneous and each basis relation is the set of arcs of the Cayley graph , where is the neighborhood of the identity of in the relation . In particular, can be treared as a color graph , where the classes of the Cayley partition are the neighborhoods of the identity of in the basis relations of .
The class of Cayley schemes is closed with respect to taking restrictions and quotients. Namely, if is a Cayley scheme over a group and is an equivalence relation, then the class of containing the identity of is a subgroup of . Moreover, the set of classes of coincides with the right -cosets of . It follows that
[TABLE]
with
[TABLE]
where in the latter case, is a normal subgroup of and is the canonical epimorphism.
Assume that the Cayley scheme is the wreath product with respect to an equivalence relation . Then for any two classes , there exists a permutation taking to ; set to be the restriction of to . Since all these are automorphisms of , the family satisfies conditions (5).
The Cayley scheme is said to be central if which is by definition of a Cayley scheme is equivalent to . One can see that is central if and only if the colored Cayley graph associated with is central.
2.5. Partial order and the WL-algorithm.
There is a natural partial order on the set of all coherent configurations on the same set . Namely, given two coherent configurations and , we set
[TABLE]
The minimal and maximal elements with respect to this order are, respectively, the trivial and discrete coherent configurations. The first one is a unique coherent configuration with at most two basis relations: and its complement to (if consists of at least two points). Every basis relation of the discrete configuration is a singleton. With respect to this order, the direct sum is the minimal coherent configuration on , the restrictions of which to and are equal to and , respectively.
One can prove that given a set , there exists a unique minimal coherent configuration such that every relation of is the union of some basis relations of . This coherent configuration is called the coherent closure of and can be constructed by the well-known Weisfeiler-Leman algorithm (WL-algorithm) [15, Section B] in time polynomial in sizes of and . To stress this fact, the coherent closure of is denoted by . For a color graph with the set of color classes, we set
[TABLE]
and write instead of . It is important to note that the automorphism group of the coherent configuration is equal to the subgroup of leaving each relation of fixed (as a set). This implies that if is a Cayley graph over , then the coherent configuration is a Cayley scheme over . Since any coherent configuration can be considered as a color graph, we extend our notation to write . Concerning the following statement, we refer to [14, Theorem 2.4].
Theorem 2.1**.**
Let and be -sets of binary relations on an -element set. Then given a bijection , one can check in time whether or not there exists an algebraic isomorphism such that . Moreover, if does exist, then it can be found within the same time.
3. Almost simple groups
In this section, we collect several known facts on finite almost simple groups and deduce some auxiliary results to be used throughout the paper.
Lemma 3.1**.**
Let be an almost simple group of order . Then
- (i)
, where , 2. (ii)
* for every containing as a normal regular subgroup.*
Proof. From the description of the automorphism groups of simple groups (see, e.g., [4, Introduction]), it follows that . Therefore, statement (i) is a consequence of the inclusions . The inclusions also imply that . Since the centralizer of in is of order [16, Exercise 4.5’], we have
[TABLE]
On the other hand, the group is -generated [5]. Thus, the number of regular subgroups of isomorphic to is at most and statement (ii) follows.
In the following statement, we use the classification of regular almost simple subgroups of a primitive group [11, Theorem 1.4].
Lemma 3.2**.**
Let be an almost simple group and . Suppose that is primitive. Then one of the following holds:
- (i)
, 2. (ii)
* and ,* 3. (iii)
* and is -transitive.*
Proof. Without loss of generality, we assume that neither , nor and . Then by aforementioned classification, exactly one of the following pairs occurs:
- (a)
, where is prime, 2. (b)
or , where is prime, 3. (c)
the twelve pairs in the table below.
The assumption , in particular, implies that . By straightforward check this excludes cases (a) with , (b) with , and cases 11 and 12 from the table. Similarly, the remaining cases in (a) and (b) as well as cases 1–4, and 9 are impossible because must divide .
In cases 5, 6, 7, and 10 from the table, we check the maximal subgroups of and show that none of them includes the subgroup isomorphic to . Indeed, in cases 5 and 6 none of the maximal subgroups contains [2, Tables 8.14, 8.28, 8.29]. In case 10, information from [11, Table 2] shows that is an extension of by a field automorphism. This group contains the only (up-to conjugation) maximal subgroup with section isomorphic to [2, Table 8.50], but the order of this subgroup is less than . In case 7, we make use of [11, Table 2] to see that . Again this group includes up to conjugation the only maximal subgroup with section isomorphic to (see [2, Table 8.50]). However, but is not isomorphic to .
This leaves us with case 8 of the table where and we arrive at case (iii) of the conclusion of the lemma.
4. The structure of automorphism groups: the principal section
4.1. Preliminaries.
Let be a finite group. The automorphism group of every central Cayley graph over contains a subgroup (see Notation). In this section, we establish some basic facts on the permutation groups satisfying the following condition:
[TABLE]
where is an almost simple group. We use a concept of the generalized wreath product of permutation groups introduced and studied in [8]. Namely, a transitive group is the generalized wreath product if it has two imprimitivity systems and such that every block of is contained in a block of and
[TABLE]
The generalized wreath product is said to be trivial if either consists of singletons or . When , the group is permutation isomorphic to the wreath product in imprimitive action, where .
Theorem 4.1**.**
Let be an almost simple group, and let be a -closed group satisfying condition (6). Then one of the following statements holds:
- (i)
* or ,* 2. (ii)
* is a nontrivial generalized wreath product.*
The proof of Theorem 4.1 is given in the end of Section 6.
4.2. The minimal block.
Let satisfy condition (6) and a -block containing the identity of . Since is a permutation group on that includes , the block is a subgroup of [16, Theorem 24.12]. Taking into account that also lies in , we conclude that is normal. Denote by the intersection of all non-singleton -blocks containing the identity of the group . Then is a -block and we call it the minimal block of .
Lemma 4.2**.**
Let satisfy condition (6) and the minimal block of . Then is normal subgroup of including . In particular, is an almost simple group such that .
Proof. According to the above remark, every -block containing the identity of is a normal subgroup of . If the block is not a singleton, then this normal subgroup is nontrivial and hence contains , because the group is almost simple. Thus, the minimal block being the intersection of nontrivial normal subgroups of is a normal subgroup and contains .
Denote by the imprimitivity system containing . Obviously,
[TABLE]
Recall that according to the definition, is a normal subgroup of leaving each block of fixed (as a set), and given , the group is induced by right multiplications of .
Lemma 4.3**.**
For any , the group is primitive and contains .
Proof. The normality of in implies that the orbits of the action of on coincide with the -cosets. It follows that the permutation group induced by this action is contained in . This proves the second statement. To prove the first statement, in view of the transitivity of , we may assume that .
Assume on the contrary that the group is not primitive. Then there exists a minimal non-singleton -block . Taking into account that , we conclude that is a also -block for every [16, Proposition 6.2]. The imprimitivity system of the group that contains coincides with the imprimitivity system containing , for otherwise by the minimality of and Lemma 4.2 applied for the group , one can choose the block so that
[TABLE]
a contradiction. Thus, is an imprimitivity system of the group . Consequently, is a non-singleton -block strictly contained in , which is impossible by the definition of .
4.3. The wreath decomposition of .
For every two sets , we write if the restriction epimorphisms
[TABLE]
are isomorphisms. In particular, the groups and are isomorphic. It is easily seen that is an equivalence relation on . This relation is -invariant, because is a normal subgroup of . Denote by the union of -cosets belonging to the class of that contains . Then is obviously a -block and hence is a normal subgroup of . Thus,
[TABLE]
The imprimitivity system of the group that contains the block is denoted by . We say that is the principal -section of , and and are the associated partitions.
Theorem 4.4**.**
Let be an almost simple group, a -closed group satisfying condition (6), and , are the partitions associated with the principal -section. Then the coherent configuration is the direct sum of the coherent configurations , where . In particular,
[TABLE]
i.e., is generalized wreath product.
Proof. The subgroup of -closed group is -closed too (see Subsection 2.2). Therefore, equality (8) follows from the first statement of the theorem, because the automorphism group of the direct sum equals the direct product of the summands (see Subsection 2.3). To prove the first statement, it suffices to verify that given ,
[TABLE]
To this end, we note that the group is the subdirect product of the transitive constituents and . Therefore, there exist uniquely determined normal subgroups and of and , respectively, and a group isomorphism such that
[TABLE]
Now if , then at least one of the epimorphisms (7) is not an isomorphism. Therefore, one of the groups and , say , is nontrivial. It follows that being a normal subgroup of the primitive group (Lemma 4.3) acts transitively on . By (10), this implies that contains the subgroup . Thus,
[TABLE]
for all and hence the group is transitive on the set .
5. The normalizer of in
The goal of this section is to prove the following theorem that shows (as we will see) that the case is very similar to the case where the group is primitive.
Theorem 5.1**.**
Let be an almost simple group, , and . Then
[TABLE]
Clearly, is a proper subgroup of and condition (6) is satisfied for . In particular, the minimal block of coincides with .
Lemma 5.2**.**
In the above notation, .
Proof. Set . Then obviously, . Since is transitive and is normal in , the orbits of form an imprimitivity system of . Denote by the block of this system that contains the identity of . We may assume that the block is not a singleton, for otherwise and we are done. Then , because is the minimal -block. Since is transitive on and is a block of , there exists such that and . However, the latter is impossible, because centralizes the subgroups and .
Proof of Theorem 5.1. By Lemma 5.2, there exists a monomorphism from to . Since the latter is isomorphic to the wreath product , this monomorphism induces a monomorphism
[TABLE]
Clearly, can be chosen so that the subgroups and of the group go to the subgroup of the group and to the subgroup of the group , respectively. Set
[TABLE]
Then since the index of in equals and is not contained in , we conclude that
[TABLE]
Note that the centralizer of the group in the group is contained in , because the group is almost simple with the socle . Since the group centralizes , this proves the first of the two following inclusions (the second one can be proved in a similar way):
[TABLE]
The first inclusion implies that . The reverse inclusion follows from the fact that the centralizer of in is equal to [16, Proposition 4.3]. Thus, we obtain the equalities:
[TABLE]
This immediately implies that and are normal in . This group has trivial center and hence can be identified with a subgroup of the direct product of the groups and (isomorphic to ). It follows that
[TABLE]
where and . Moreover in view of formulas (13), the group intersects each of the groups and trivially. Therefore,
[TABLE]
Now, the inclusion and formula (14) show that . Formula (12) yields that . Since the permutation lies in , formula (11) holds.
6. Symmetric and normal types of the automorphism group
Let be an almost simple group, be a -closed group satisfying condition (6), and the principal -section of . Then the group is almost simple (Lemma 4.2) and is primitive (Lemma 4.3). We say that is of symmetric type if either , or and is -transitive; if and , the group is said to be of normal type. The following statement is a straightforward consequence of Lemma 3.2 and the above definitions.
Proposition 6.1**.**
Let be an almost simple group, and let be a -closed group containing . Then is of symmetric or normal type.
Let us study a group of symmetric and normal types in detail. As the following statement shows, any group of symmetric type is, in fact, the wreath product in imprimitive action. In what follows, and are the partitions associated with the principal -section of .
Theorem 6.2**.**
Let be a group of symmetric type. Then
[TABLE]
In particular, is permutation isomorphic to the wreath product in imprimitive action.
Proof. Let and . We claim that there exists a bijection , for which
[TABLE]
where is the graph of . Indeed, consider the group . Since , the group acts faithfully on and . As the group is of symmetric type, each of these actions is 2-transitive. However, has a unique faithful 2-transitive representation of degree : this is obvious if and follows from the classification of 2-transitive groups if (see, e.g., [3, Table 7.4]). Consequently, among point stabilizers , , there are exactly distinct, and also whenever and are distinct points in . Therefore, for every there is the only such that
[TABLE]
Thus, the required bijection takes to .
To prove that , assume on the contrary that contains a block other than . Denote by the bijection defined in the above claim for and . Then by the assumption, the element , where is the identity of the group , does not belong to . The binary relation is invariant with respect to the group and hence for all ,
[TABLE]
where and are the permutations of induced by the right and left multiplication by . This implies that the element centralizes . However, this is impossible, because and the group is almost simple.
Let us prove the second equality. The -closedness of implies that is -closed. Furthermore, in view of the -transitivity of the group its the -closure equals . By Theorem 4.4 and equality , this implies that
[TABLE]
Thus, for all and we are done.
Theorem 6.2 shows that in the case of symmetric type, the group is the largest possible. In the normal type case, the group is quite small. More exactly, the following statement holds.
Theorem 6.3**.**
Let be a group of normal type. Then .
Proof. By the hypothesis of the theorem, and . By Lemma 4.2, the first equality implies that and hence
[TABLE]
The second inclusion implies that is a characteristic subgroup of the group for all . By the definition of , this implies that is a characteristic subgroup of the group . However, the latter group is normal in . Thus,
[TABLE]
It follows that is contained in the normalizer of in . However, this normalizer is contained in by Theorem 5.1 applied for with taking into account equality (16).
Proof of Theorem 4.1. Let be the principal -section of the group . By Proposition 6.1, the group is of symmetric or normal type. Suppose first that . Then statement (i) of Theorem 4.1 holds. Indeed, if is of symmetric type, then (Theorem 6.2), whereas if is of normal type, then (Theorem 6.3) and the required statement follows from the fact that is normal in . Finally, if , then statement (ii) of Theorem 4.1 holds by Theorem 4.4.
Proof of Theorem 1.3. First, assume that the group is a nontrivial generalized wreath product. Note that is a unique proper normal subgroup in and . Therefore the generalized wreath product must be a usual one and the group is permutation isomorphic to the wreath product in imprimitive action for some group . It follows that is an orbit of the point stabilizer , where is the identity of . Since is a union of some orbits of , we conclude that , a contradiction.
The group is 2-closed as the automorphism group of a graph. The normality of implies that satisfies condition (6) with for . Finally, is not a nontrivial generalized wreath product by above, and , because the graph is neither complete nor empty. Thus, by Theorem 4.1, the group is normal in and
[TABLE]
hence or , because . However, since is a normal subset of a symmetric group, we have , so the graph has the automorphism , . Thus, , as required.
7. Finding the principal section in a Cayley scheme
7.1. The main resut.
Let be a central Cayley scheme over an almost simple group . Then the group is -closed and satisfies condition (6). Therefore, is of symmetric or normal type by Proposition 6.1. In this section, we develop an algorithmic technique to determine (with no in hand) which of these cases occurs for the scheme . The main result here is Theorem 7.1 below which immediately follows from Corollaries 7.3 and 7.5 proved in Subsections 7.2 and 7.3, respectively.
Theorem 7.1**.**
Given a central Cayley scheme over an almost simple group of order , one can determine the type of and find the principal -section of in time .
7.2. The case of symmetric type.
For a group , denote by and the partition of into the right -cosets and the Cayley scheme with , respectively. Recall that is the trivial coherent configuration on . Denote by the set of groups such that and
[TABLE]
Lemma 7.2**.**
In the above notation, the following statements hold:
- (i)
if the group is of symmetric type, then the set is nonempty and the minimal block of is the largest (by inclusion) element of , 2. (ii)
* is of symmetric type if and only if contains ,*
Proof. To prove statement (i), assume that the group is of symmetric type. Set and to be the partitions associated with the principal -section of . Then by Theorem 6.2, we have and for all . By Theorem 4.4, this implies that
[TABLE]
The minimality of the direct sum implies that , which proves formula (17) for and , in particular, is nonempty. If is not the largest element of , then there exists such that . It follows that
[TABLE]
where is the identity subgroup of . Hence there is a permutation that moves the identity of to and leaves all non-identity elements of fixed. But this is impossible, because is a -block.
To prove the necessity for statement (ii), let be of symmetric type. Then formula (18) holds. Therefore, if is the partition of into the cosets of , then refines and hence
[TABLE]
where is the partition of induced by . Consequently, . Conversely, assume on the contrary that is of normal type. Then by Theorem 6.3. Therefore . On the other hand, since , we have , a contradiction.
From statement (i) of Lemma 3.1, it follows that the number of groups containing is at most , where is the order of ; in particular, . Moreover, for each , the coherent configuration can be efficiently found by the WL-algorithm and condition (17) can be verified by checking at most basis relations. Therefore, the set can be found in time . By statement (ii) of Lemma 7.2, this is enough to test efficiently whether or not is of symmetric type, and if it so, then to find the minimal block of the group (statement (i) of the same lemma).
Corollary 7.3**.**
Given a central Cayley scheme over an almost simple group of order , one can test in time whether the group is of symmetric type, and (if so) find the principal section of within the same time.
7.3. The case of normal type.
In view of Corollary 7.3 and Proposition 6.1, one can efficiently test whether the automorphism group of a central Cayley scheme is of normal type. Denote by the set of all groups such that and
[TABLE]
where the left-hand side denotes the subgroup of that leaves each point of fixed and coincides with on .
Lemma 7.4**.**
Suppose that the group is of normal type and is the principal -section of . Then , the set is nonempty, and is the smallest element of .
Proof. Lemma 3.2 yields that . From Theorem 4.4, it is easily follows that . Assume on the contrary that the group is not the smallest in . Then there is a group such that . Take a non-identity element and denote by (respectively, ) the permutation on acting on (respectively, ) by right multiplication by and acting trivially outside (respectively, ). Then , the permutation
[TABLE]
is not identity on , and Clearly, belongs to , and hence to because is of normal type. However, as is easily seen, the identity element is the only element of that leaves all points of fixed, a contradiction.
Again from statement (i) of Lemma 3.1 it follows that the number of groups such that is at most , and so is . For every and each one can efficiently test whether is an automorphism of . Thus, Lemma 7.4 immediately implies the following statement.
Corollary 7.5**.**
Given a central Cayley scheme over an almost simple group of order , one can test in time whether the group is of normal type, and (if so) find the principal section of within the same time.
8. A majorant for the coset of isomorphisms
Throughout this section, we assume that is a central Cayley scheme over an almost simple group and . The principal -section of and the associated partitions are denoted by and and , respectively. The equivalence relations corresponding to the partitions and , are denoted by and . Let be an algebraic isomorphism from onto a Cayley scheme over an almost simple group . Assume that
[TABLE]
where in what follows, the group , the principal section , the partitions and , and the equivalence relations and are defined for the scheme in a similar way.
Lemma 8.1**.**
In the above notation, and . Moreover, the groups and either both of symmetric type, or both of normal type.
Proof. The first statement follows from assumption (20). To prove the second one, we note that by Lemma 7.2 the group is of symmetric type if and only if the scheme is isomorphic to the wreath product , where is the quotient of modulo the equivalence relation . Since algebraic isomorphisms respect wreath products, we are done.
For all and , the algebraic isomorphism induces an algebraic isomorphism
[TABLE]
that takes a relation to the relation for all basis relations of the scheme , where . It follows that if and is trivial, then is trivial for all .
For each , set
[TABLE]
where, for brevity, denotes the restriction of the permutation group to the set . Note that the form of the group does not depend on , and contains for all (Theorems 6.2 and 6.3). Furthermore, if , then any permutation of taking to induces a permutation isomorphism from the group onto the group .
Lemma 8.2**.**
* for all and all .*
Proof. Without loss of generality, we may assume that both and are of normal types (Lemma 8.1). Let and . Then in view of (20),
[TABLE]
where . Thus, it suffices to verify that , where and . However, by the definition of , we have for suitable and
[TABLE]
where is the bijection induced by right multiplication by . A similar statement holds for , , and a suitable bijection . Therefore, without loss of generality, we may assume that and .
Recall that is an almost simple group with , and also . Therefore, . The same is true with , , and , replaced by , , and , respectively. Taking into account that takes to , we conclude that
[TABLE]
This implies that takes the normalizer of the group to the normalizer of the group . However by Theorem 5.1, these normalizers are equal to and . Thus, takes the first of these groups to the second, and we are done.
Definition 8.3**.**
Denote by the set of all bijections taking to and satisfying the following conditions for every :
[TABLE]
Let us find the explicit form of the set when and . In this case, is obviously a subgroup of preserving the partition . Condition (22) means that belongs to the intersection of and the normalizer of in for all . This proves the first of the equalities
[TABLE]
the second equality follows from the fact that any induces the permutation isomorphism from onto that induces . Thus, the group is permutation isomorphic to the wreath product (in imprimitive action). For arbitrary and , an for each , we obviously have
[TABLE]
Thus if the set is not empty, then it can be given by a generator set of the group and the bijection . In the sense of the following statement, the set can be called a majorant of .
Theorem 8.4**.**
. Moreover, the set can be found in time .
Proof. The first statement immediately follows from Lemma 8.2. To prove the second one, it suffices to find the set
[TABLE]
where . Indeed, if this set is empty, then obviously so is the majorant . On the other hand, if , then to construct the majorant given by formula (23) it suffices to find and the bijection defined as follows:
[TABLE]
where is an arbitrary bijection from onto taking to , and the bijections and are induced by the right multiplications by the elements and such that and .
To find the sets and , assume first that is of symmetric type (recall that this can efficiently checked by Theorem 7.1). Then the coherent configurations and are trivial. Thus, and for any bijection ,
[TABLE]
Let now be of normal type. Then and . In particular, can be found in time . Furthermore, every element takes to (Lemma 8.2), and induces a permutation group isomorphism from onto a group
[TABLE]
By statement (ii) of Lemma 3.1, the set is of cardinality at most for some constant , and all its elements can be found by exhaustive search of all -generated subgroups of the group . Since for a fixed , there are at most distinct elements taking to , one can test in time , whether the set is not empty and (if so) find it in the form
[TABLE]
with arbitrary .
9. Proof of Theorem 1.1
9.1. Reduction to Cayley schemes.
Let be a central Cayley graph over an almost simple group , the set of color classes of , and . The principal -section of and the associated partitions are denoted by and and , respectively. Set
[TABLE]
where and are the equivalence relations corresponding to the partitions and . For other central Cayley graphs , we use similar notation, e.g., and denote the underlying group and the automorphism group of , respectively.
Lemma 9.1**.**
Given central Cayley graphs and over almost simple groups and , respectively, one can construct in time the Cayley schemes and over the same underlying groups and check whether there exists a (unique) algebraic isomorphism such that
[TABLE]
and (if so) find within the same time. Moreover, , , and also
[TABLE]
Proof. By Theorem 7.1, the principal sections and hence the equivalence relations , and , can be found in time . Therefore the first part of the statement immediately follows from Theorem 2.1. To prove the second one, we observe that every takes the group to the group . By the definition of the minimal block (Subsection 4.2) this implies that takes to and hence takes to . Thus, the isomorphism induces . This means that and hence the left-hand side of (25) is contained in the right-hand side. Since the reverse inclusion is obvious, equality (25) is completely proved. Next, if , then and equality (25) shows that . Similarly, .
9.2. Determining the coset of isomorphisms.
Denote by the canonical epimorphism from onto . Then induces a map taking the set of basis relations of the Cayley scheme over to the set of basis relations of the quotient Cayley scheme over . In particular, takes to . Set to be the Cayley graph over with color classes , . For any set of the bijections taking to and to , we denote by the set of bijections induced by .
Theorem 9.2**.**
Let and be central Cayley graphs over almost simple groups and , respectively. Assume that the algebraic isomorphism from Theorem 9.1 does exist. Then
[TABLE]
where , , and the right-hand side consists of all for which .
Proof. To prove that that the left-hand side of (26) is contained in the right-hand side, let . Then the uniqueness of the principal sections implies that
[TABLE]
Therefore, the isomorphism induces the algebraic isomorphism . By Theorem 8.4, this implies that . Consequently, the induced bijection belongs to the set . Since obviously , we conclude that belongs to the right-hand side of (26), as required.
Conversely, let belong to the right-hand side of (26). By formula (25) in Lemma 9.1, it suffices to verify that induces . To this end, let . Assume first that . Then equals the union of , . Therefore since , conditions (22) are satisfied for all and hence
[TABLE]
Now assume that is outside the equivalence relation . Let us prove that
[TABLE]
where and . Since , the relation is outside the equivalence relation . Therefore, it suffices to verify the first equality of (27); denote the right-hand side of this equality by . Clearly, . Conversely, let for some . Since is outside , we conclude that is a basis relation of the coherent configuration (Theorem 4.4). Since , it follows that this basis relation is contained in a basis relation of which equals , because . Therefore, . This completes the proof of (27) implying and .
On the other hand, the graph isomorphism induces an algebraic isomorphism that coincides with the restriction of the algebraic isomorphism modulo . Thus,
[TABLE]
as required.
9.3. The algorithm.
In the algorithm below, the input is given by two central Cayley graphs and over almost simple groups and , respectively. It is assumed that these groups are presented by the multiplication tables. The output consists of the set , which is either empty or equals the set for some . Here, the group is presented by a generating set.
Central Cayley graph isomorphism test
Step 1. Find the principal sections of the automorphism groups of the (central Cayley) schemes and (Theorem 7.1); denote by , and , the associated partitions of and , respectively.
Step 2. Find the schemes and and the algebraic isomorphism satisfying condition (24); if does not exist, output .
Step 3. Find the set (Theorem 8.4).
Step 4. Using the graph isomorphism and coset intersection algorithms from [1], find the set and then the set .
Step 5. Output .
To complete the proof of Theorem 1.1, we show that the above algorithm correctly finds the set in time . Note that every graph isomorphism induces an algebraic isomorphism satisfying condition (24). Therefore, the output at Step 2 is correct. Thus, the correctness of the output at Step 5 and hence of the algorithm immediately follows from Theorem 9.2.
To estimate the running time, we note that all the steps except for Step 4 run in polynomial time (Theorem 7.1, Lemma 9.1, and Theorem 8.4). Furthermore, the graph isomorphism and coset intersection algorithms from [1] are applied at Step 4 to graphs with vertices and to the cosets contained in , respectively. Each of these algorithms runs in time at most . Since , the complexity of this step does not exceed
[TABLE]
for sufficiently large and a suitable constant . Thus, the running time of the algorithm is polynomial in , as required.
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