Group isomorphism is nearly-linear time for most orders
aa r X i v : . [ c s . CC ] D ec GROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS
HEIKO DIETRICH AND JAMES B. WILSON
Abstract.
We show that there is a dense set Υ ⊆ N of group orders and a constant c such that forevery n ∈ Υ we can decide in time O ( n (log n ) c ) whether two n × n multiplication tables describeisomorphic groups of order n . This improves significantly over the general n O (log n ) -time complexityand shows that group isomorphism can be tested efficiently for almost all group orders n . We alsoshow that in time O ( n (log n ) c ) it can be decided whether an n × n multiplication table describesa group; this improves over the known O ( n ) complexity. Introduction
Given a natural number n , there are many structures that can be recorded by an n × n table T taking values T ij in [ n ] = { , . . . , n } . Isomorphisms of these tables are permutations σ on [ n ] with T σ ( i ) σ ( j ) = σ ( T ij ) for all i, j ∈ [ n ]. It is convenient to assign these tables either a geometric oralgebraic interpretation. A geometric view treats these as edge colored directed graphs or as therelations of an incidence structure. We will consider the algebraic interpretation where the tabledescribes a binary product ∗ : [ n ] × [ n ] → [ n ].The upper bound complexity to decide isomorphism is O ( n !) by testing all permutations. Bet-ter timings arise when we consider subfamilies of structures, for example, by imposing equa-tional laws on the product such as associativity a ∗ ( b ∗ c ) = ( a ∗ b ) ∗ c (i.e. semigroups ) orthe existence of left and right fractions (i.e. quasigroups or latin squares ). Booth [20, p. 132]observed that the complexity of isomorphism testing of semigroups is polynomial-time equivalentto the complexity of graph isomorphism. At the time, that complexity was subexponential, butit has since been shown by Babai [1] to be in quasi-polynomial time with a highly inventive al-gorithm. Meanwhile Miller [19, 20] observed that the complexity of quasigroup isomorphism is inquasi-polynomial time n O (log n ) through an almost brute-force algorithm: since quasigroups of order n are generated by log n elements, a brute-force comparison of all ⌈ log n ⌉ -tuples either finds anisomorphism between two quasigroups or determines that no isomorphism exists.An intriguing bottleneck to further improvement has been the case of groups that have associativeproducts with an identity and left and right inverses. Because these are quasigroups, they havea brute-force isomorphism test with complexity n O (log n ) ; Miller gave credit to Tarjan for thiscomplexity. Guralnick and Lucchini [4, Theorem 16.6] showed independently that every finitegroup of order n can be generated by at most d ( n ) elements and d ( n ) µ ( n ) + 1, where µ ( n ) isthe largest exponent of a prime power divisor p µ ( n ) of n . Thus, the complexity of brute-force groupisomorphism testing is more accurately described as n O ( µ ( n )) . Since µ ( n ) ∈ Θ(log n ) when n = 2 ℓ m with m ∈ O (1), this does not improve on the generic n O (log n ) bound. Group isomorphism testingseems to be a leading bottleneck to improving the complexity of graph isomorphism, see Babai [1,Section 13.2]. Even so, we prove here that for most orders, group isomorphism is in nearly-lineartime compared to the input size; this also shows that groups of general orders are not a bottleneckto graph isomorphism. School of Mathematics, Monash University, AustraliaDepartment of Mathematics, Colorado State University, Colorado, USA
E-mail addresses : [email protected], [email protected] . Key words and phrases. group isomorphism, complexity.
Current state of group isomorphism.
Surprisingly, the brute-force n O (log n ) complexity ofgroup isomorphism has been resilient. Progress has fragmented into work on numerous subclasses X of groups; the precise problem studied today is: X -GroupIso Given a pair (
T, T ′ ) of n × n tables with entries in [ n ] representing groups in X , Decide if the groups are isomorphic.Grochow-Qiao [13] give a detailed survey of recent progress; here we summarize a few results relatedto our setting. Iliopoulos, Karagiorgos-Poulakis, Vikas, and Kavitha [15] progressively improvedthe complexity for the class A of abelian groups (where the product satisfies a ∗ b = b ∗ a for all a, b ),resulting in a linear-time algorithm for A - GroupIso in a RAM model (more on this below).Wagner-Rosenbaum [25] gave an n .
25 log n + O (1) time algorithm for the class N p of groups of ordera power of a prime p , and later generalized this to the class of solvable groups. Li-Qiao [17] provedan average run time of n O (1) for an essentially dense subclass N p, ⊂ N p . Babai-Codenotti-Qiao [2]proved an n O (1) bound for the class T of groups with no nontrivial abelian normal subgroups.Das-Sharma [6] described a nearly-linear time algorithm for the class of Hamiltonian groups, againin a RAM model.Other research combines results for separate classes by considering isomorphism between groupsthat decompose into a subgroup in class X and a quotient in class Y , see also Section 2; we callthis the ( X , Y ) -extension problem . Le Gall [16] studied ( A , C )-extensions, where C consists of cyclicgroups. Grochow-Qiao [13] considered ( A , T )-extensions, and outlined a general framework forsolving extension problems.A further class of algorithms considers terse input models, such as black-box models or groupsof matrices or permutations; we refer to Seress [26, Section 2] for details of those models. In thisformat, groups can be exponentially larger than the data it takes to specify the group. Using thismodel, the second author proved an (log n ) O (1) -time algorithm for subgroups and quotients of finiteHeisenberg groups, and further variations in collaborations with Lewis and Brooksbank-Maglione;see [29] and the references therein. Recently, in [8] the authors proved a polynomial-time isomor-phism test for permutation groups of square-free and cube-free orders. These examples demonstratethat input models may have an outsized influence on the complexity of group isomorphism.Some of the motivation of this and earlier work [8] has been the observation that, in contrastto graph isomorphism, the difficulty of group isomorphism is influenced by the prime power fac-torization of the group orders n . For example, if n = 2 e ± n and isomorphism can be tested by comparing orders.Yet, there are n e / − O ( e ) isomorphism types of groups of prime power order n ∓ e , see [4,p. 23]. As of today, isomorphism testing of groups of order 2 e has the worst-case complexity.1.2. Main results.
The main result of this paper is a proof that group isomorphism can be testedefficiently for almost all group orders n in time O ( n (log n ) c ) for some constant c , if the groupsare input by their Cayley tables, that is, by n × n tables describing their multiplication maps[ n ] × [ n ] → [ n ]. To make “almost all” specific, we define the density of a set Ω ⊆ N to be the limit δ (Ω) = lim n →∞ | Ω ∩ [ n ] | /n ; the set Ω is dense if δ (Ω) = 1. By abuse of notation, Ω- GroupIso denotes the isomorphism problem for the class of groups whose orders lie in Ω. All our complexitiesare stated for deterministic Turing machines; we give details in Section 2.
Theorem 1.1.
There is a dense subset Υ ⊂ N and a deterministic Turing machine that decides Υ - GroupIso for n ∈ Υ in time O ( n (log n ) c ) for some constant c . We provide a proof in Section 4.1. Since every multiplication table [ n ] × [ n ] → [ n ] can be encodedand recognized from a binary string of length Θ( n log n ), the algorithm of Theorem 1.1 is nearly-linear time in the input size. The dense set Υ is specified in Definition 2.3; here we remark that ROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS 3 we can determine in time O ( n ) whether n ∈ Υ, and the complexity for brute-force isomorphismtesting of groups of order n ∈ Υ is n O (log log n ) . Because of this, we would have been content witha polynomial-time bound; being able to prove nearly-linear time bound was a surprise.Our set Υ excludes an important but difficult class of group orders, specifically orders that havea large power of a prime as a divisor. Theorem 1.1 therefore goes some way towards confirming theexpectation that groups of prime power order are the essential bottleneck to group isomorphismtesting. Indeed, examples such as provided in [29] show that large numbers of groups of primepower order can appear identical and yet be pairwise non-isomorphic. In fact, known estimates onthe proportions of groups show that most isomorphism types of groups accumulate around orderswith large prime powers, see [4, pp. 1–2]. So our Theorem 1.1 should not be misunderstood assaying that group isomorphism is efficient on most groups, just on most orders. Even so, we seein results like Li-Qiao [17] and Theorem 1.1 the beginnings of an approach to show that groupisomorphism is polynomial-time on average, and we encourage work in this direction.The solutions of X - GroupIso cited so far deal with the problem in the promise polynomialhierarchy [12] where one promises that inputs are known to be groups and that they lie in X . Torelate those solutions to the the usual deterministic polynomial-time hierarchy forces us to considerthe complexity of the associated membership problem: X -Group Given a binary string T , Decide if T encodes the Cayley table of a group contained in X .While it is straight-forward to verify that an input encodes an n × n Latin square (i.e. quasigroup),current methods available in the literature seem to require O ( n ) steps to verify that the productis associative, see [11, Chapter 2]. Here we present an improvement that solves G - Group for theclass G of all finite groups in time O ( n (log n ) d ) for some constant d . We note that Rajagopalan-Schulman [23, Theorem 5.2] provide an O ( n log n ) algorithm for this task, but they cost the binaryoperation as O (1), which gives an upper bound of O ( n (log n ) ) on a Turing machine model. Theorem 1.2.
There is a deterministic Turing machine that decides in time O ( n (log n ) d ) forsome constant d whether a multiplication table on [ n ] describes a group and, if so, returns a homo-morphism [ n ] → Sym n into the group Sym n of permutations on [ n ] . We prove Theorem 1.2 in Appendix 4.2. Thus, we may cast Theorem 1.1 as nearly-linear timein the deterministic polynomial-time hierarchy, that is, it properly accepts or rejects all stringswithout assuming external promises on these inputs. Theorem 1.2 also offers a hint that our strategypartly entails working with data structures for permutation groups, instead of working with themultiplication tables directly. This is responsible for much of the nearly-linear time complexity ofthe various group theoretic routines upon which we build our algorithm for Theorem 1.1.While we provide a self-contained proof, Theorem 1.2 is an example of a general approach weare developing for shifting promise problems to deterministic problems, see also Section 5. Promiseproblems are especially common whenever inputs are given by compact encodings such as black-boxinputs, see Goldreich [12]. In ongoing work [9], we introduce a more general process for verifyingpromises by specifying inputs not as strings for a Turing machine, but rather as types in a sufficientlystrong Type Theory. Theorem 1.2 can be interpreted as an example of such an input where therows of the multiplication table are themselves treated as inhabitants of a permutation type. Thealgorithm then effectively type-checks that these rows satisfy the required introduction rules for apermutation group type. Type-checking is not in general decidable so the effort is to confirm anefficient complexity for specific settings. As a by-product of such models, the subsequent algorithmsalso profit from using these rich data types; for more details on this topic we refer to [9].1.3.
Structure.
In Section 2 we introduce relevant notation and state Theorems 2.4–2.6, which arethe main ingredients for our proof of Theorem 1.1. Since the proofs of Theorems 2.4–2.6 are more
GROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS involved and partly depend on technical Group Theory results, we delay them until Appendix A.In Section 3 we discuss some algorithmic results required for our proof of Theorem 2.6. Proofs forour main results are provided in Section 4. A conclusion and outlook are given Section 5.2.
Notation and preliminary results
Computational model.
Throughout n is the order of the multiplication tables used as inputto programs, so input lengths are Θ( n log n ). We write ˜ O ( n d ) for O ( n d (log n ) c ), where c and d are constants, and we note that O ( n (log n ) O ((log log n ) c ) ) ⊂ O ( n ε ) for any ε >
0; below we use ε = 0 . O ( n log n ) to storeour associated permutation group representations developed in the course of Theorem 1.2. In thismodel, carrying out a group multiplication requires one to reposition the tape head to the correctproduct, at the cost of O ( n log n ). That will be prohibitive for our given timing, so our first orderof business will be to replace the input with an efficient ˜ O ( n )-time multiplication, cf. Remark 4.1.For comparison, work of Vikas and Kavitha [15] provide isomorphism tests for abelian groups using O ( n ) group operations. Such an algorithm can be considered as linear time in a random accessmemory (RAM) model where group and arithmetic operations are stated as a unit cost. This ispartly how it is possible to produce a running time shorter than the input length. In general,an f ( n )-time algorithm in a RAM model produces an O ( f ( n ) )-time algorithm on a TM, see [22,Section 2], although a lower complexity reduction may exist for specific programs. In particularresults in [6] [15] are no more than ˜ O ( n )-time on a Turing Machine.We note that if we would work in the promise hierarchy and use a RAM model (that is, inputtables are pre-loaded in registers and group multiplication is costed O (1)), then our algorithm fordeciding isomorphism of groups of order n ∈ Υ requires time O ( n . ).2.2. Group theory preliminaries.
We follow most common conventions in group theory, e.g. asin [24, 26]. Given a group G , a subgroup H G is a nonempty subset which is a group with theinherited operations from G . Homomorphisms between groups G and K are functions f : G → K with f ( xy ) = f ( x ) f ( y ) for all x, y ∈ G . For S ⊆ G , let h S i be the intersection of all subgroupscontaining S ; it is the smallest subgroup of G containing S , also called the subgroup generated by S . The commutator of group elements x, y is [ x, y ] = x − y − xy , and conjugation is written as x y = x [ x, y ]. For X, Y ⊆ G , let [ X, Y ] = h [ x, y ] : x ∈ X, y ∈ Y i . The number of elements in G , the order of G , is denoted | G | ; in this work, G always is a finite group.Given a set π of primes, a subgroup H G is a Hall π -subgroup if for every p ∈ π dividing | G | ,we have that p divides | H | , but not the index | G : H | = | G | / | H | . If π = { p } , then H is a Sylow p -subgroup . A further convention is to let π ′ denoted the complement of π in the set of all primesand to speak of Hall π ′ -subgroups. The π -factorization of an integer n > n = ab , where everyprime divisor of b lies in π , and no prime divisor of a lies in π .A subgroup B is normal in G , denoted B E G , if [ G, B ] B . The group G is simple if its onlynormal subgroups are { } and G . A composition series for G is a series G = G > . . . > G m = { } of subgroups with each G i +1 E G i and each composition factor G i /G i +1 is simple. The group G is solvable if every composition factor is abelian.A normal subgroup B E G splits in G , denoted G = H ⋉ B , if there is H G with H ∩ B = { } and G = h H, B i . Let Aut( B ) be the set of invertible homomorphisms B → B . If G = H ⋉ B and h ∈ H , then conjugation b b h defines θ ( h ) ∈ Aut( B ), and θ : H → Aut( B ), h θ ( h ),is a homomorphism. Conversely, given ( H, B, θ ) with homomorphism θ : H → Aut( B ), there is agroup H ⋉ θ B on the set { ( h, b ) : h ∈ H, b ∈ B } with product ( h , b )( h , b ) = ( h h , b h b ); herewe abbreviate b h = θ ( h )( b ). If conjugation in G = H ⋉ B induces θ : H → Aut( B ), then G isisomorphic to H ⋉ θ B . In fact, the following observation holds; we will use this lemma only in thesituation that B and ˜ B are cyclic groups; this case of Lemma 2.1 is proved in [7, Lemma 2.8]. ROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS 5
Lemma 2.1.
Let G = H ⋉ θ B and ˜ G = ˜ H ⋉ ˜ θ ˜ B . If α : H → ˜ H and β : B → ˜ B are isomorphismssuch that for all h ∈ H ˜ θ ( α ( h )) = β ◦ θ ( h ) ◦ β − , (1) then ( h, b ) ( α ( h ) , β ( b )) is an isomorphism G → ˜ G . Conversely, if G and ˜ G are isomorphic and H and B have coprime orders, then there is an isomorphism G → ˜ G of this form. Number theory preliminaries.
Our algorithm for Theorem 1.1 depends on crucial numbertheoretic observations. For integers n , we characterize a family of prime divisors we call stronglyisolated , and we show that any group G of order n in our dense set Υ decomposes as G = H ⋉ B such that the prime factors of | B | are exactly the strongly isolated prime divisors of n that are largerthan log log n ; we also prove that B is cyclic. This reduces our isomorphism test to considering thedata ( H, B, θ ) and Lemma 2.1. Not just isomorphism testing, but many natural questions of finitegroups reduce to properties of (
H, B, θ ), and so this decomposition has interesting implications forcomputing with groups generally. Note that for a group G with order n ∈ Υ, the decompositiondescribed above defines integers a = | H | and b = | B | that depend only on n . Our definition of Υwill also imply that b is square-free, and if a prime power p e with e > n , then p e log n and p e | a . With B being cyclic, the group theory of B is elementary, and while the group theoryof H can be quite complex, we will see in Theorem 2.5 that H has relatively small size, meaningthat brute-force becomes an efficient solution. The following definition is central for our work. Definition 2.2.
Let n ∈ N . Write 2 ν ( n ) for the largest 2-power dividing n . A prime p | n is isolated if k = 0 for every prime power q k with q k | n and p | ( q k − p ∤ | T | forevery non-abelian simple group T of order dividing n , then p is strongly isolated . We write π n forthe set of strongly isolated prime divisors of n and define π big n = { p ∈ π n : p > log log n } . For example, 31 is isolated in 2 · ·
31 but not in 2 · ·
31 or in 2 · ·
31. As indicated above,the properties of our dense set Υ are a critical ingredient in our algorithm for Theorem 1.1; we arenow in the situation to give the formal definition.
Definition 2.3.
Let Υ ⊆ N be the set of all integers n that factor as n = ab such that:a) if p | a is a prime divisor, then p log log n and, if p e | a , then p e log n ;b) if p | b is a prime divisor, then p > log log n and p | n is isolated;c) the factor b is square-free, that is, if p e | b is a prime power divisor, then e ∈ { , } . Theorem 2.4.
The set Υ is a dense subset of N . A proof of Theorem 2.4 and more details on Υ are given in Appendix A.1 and Section 5.2.4.
Splitting results.
As mentioned in Section 2.3, the properties of n ∈ Υ impose limits on thestructure of groups of order n ; we prove the next theorem in Appendix A.2: Theorem 2.5.
Every group G of order n ∈ Υ has a unique Hall π big n -subgroup B , which is cyclic,and G = H ⋉ B for some subgroup H G of small order | H | ∈ (log n ) O ((log log n ) ) . Remark 2.1.
In fact, our proof of Theorem 2.5 also shows the following result for any group G of any order n ∈ N : If G is solvable and p | n is an isolated prime, or if G is non-solvable, p | n isa strongly isolated prime, and p > ν ( n ), then G has a normal Sylow p -subgroup S E G ; in bothcases, G = H ⋉ S for some H G by the Schur-Zassenhaus Theorem [24, (9.1.2)].The next theorem shows that we can construct generators for the decomposition in Theorem 2.5;we discuss the proof of Theorem 2.6 in Appendix A.2. Theorem 2.6.
There is an ˜ O ( n . ) -time algorithm that, given a group G of order n ∈ Υ , returnsgenerators for H, B G such that G = H ⋉ B and B is a Hall π big n -subgroup. GROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS Algorithmic preliminaries: presentations and complements
We assume now that our input has been pre-processed and confirmed to be a group by our algorithmfor Theorem 1.2. In so doing, we also produce a series of important data types and accompanyingroutines, including the following for each input group of order n : • a permutation group representation, • a generating set of size O (log n ), • an algorithm to write group elements as a product of the generators in time ˜ O ( n ), • an algorithm to multiply two group elements in time in (log n ) O ( n ) , and • an algorithm to test equality of elements in the group in time ˜ O ( n ).Remark 4.1 explains some of these routines in more detail. The advantage is that we can nowmultiply group elements without moving the Turing machine head over the original Cayley tableswhich would cost us ˜ O ( n ) steps for each group operation.Many of the following observations are variations on classical techniques designed originally forpermutation groups, e.g. as in [18, 26, 27]. We include proofs here to demonstrate nearly-lineartime when applied to the Cayley table model. For simplicity we assume that all generating setscontain 1. All our lists have size O ( n ), so all searches can be done in nearly-linear time.A membership test for a subgroup L G is a function that, given g ∈ G , decides whether g ∈ L .For example, a membership test for the center Z ( G ) G is to report the outcome of the testwhether g ∈ G satisfies g ∗ s = s ∗ g for all s ∈ S . This test defines Z ( G ) without having specifieda generating set for it. The next result shows that the Cayley graph of G can be computed innearly-linear time; note that Cay( S, { } ) is indeed the usual Cayley graph with respect to S . CayleyGraph
Given a group G generated by S ⊆ G and a membership test for a subgroup L , Return the Cayley graph Cay(
S, L ) = { ( xL, sxL ; s ) | s ∈ S, x ∈ G } with a spanningtree (a so-called Schreier tree ). Proposition 3.1.
Let G = h S i with | G | = n and | S | ∈ O (log n ) , and let L G be given by a mem-bership test. CayleyGraph can be solved using O ( | S || G : L | ) ⊂ ˜ O ( n ) group multiplications andmembership test applications. Once solved, there is an ˜ O ( n ) -time algorithm to compute generatorsfor L , and a ˜ O ( | G : L | ) -time algorithm that given g ∈ G finds a word ¯ g in S with ¯ gL = gL .Proof. We initialize graphs C and T , both with vertex set V = { L } and empty edge sets. Now usethe orbit-stabiliser algorithm [26, Section 4]: While there is s ∈ S and a vertex xL ∈ V such that sxL / ∈ V , add sxL to V and add ( xL, sxL ; s ) to the edge sets of C and T . Otherwise, sxL ∈ V and add ( xL, sxL ; s ) only to the edge set of C . One compares xL = yL by deciding x − y ∈ L viathe membership test. The algorithm terminates after O ( | S || G : L | ) steps, and then | V | = | G : L | and V = Cay( S, L ) with Schreier tree T . This spanning tree determines a transversal T for L in G , that is, G is the disjoint union of all tL with t ∈ T . Given g ∈ G , there is a unique vertex xL in T with gL = xL and unique labeled path L t → t L t → · · · t r → xL with each t i ∈ S , that is, ¯ gL = gL for ¯ g = t r t r − · · · t . Schreier’s lemma [26, Lemma 4.2.1] shows that L is generated by the set of all( st ) − st where s ∈ S and t ∈ T . The number of such generators is O ( | S || G : L | ) and the productscost (log n ) O (1) each. The computation of ¯ g in time ˜ O ( | G : L | ) is as described in Remark 4.1. (cid:3) Corollary 3.2.
Let G = h S i be a group of order n and let π be a set of primes. If G has a normalHall π -subgroup B , then there is an ˜ O ( n . ) -time algorithm that returns generators for B , and amembership test for B that decides in poly-logarithmic time.Proof. Since B is the unique Hall π -subgroup, g ∈ G lies in B if and only if the order of g divides b = | B | ; therefore we may test membership in B by testing if g b = 1. This can be ROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS 7 done in O (log b ) group products using fast exponentiation, followed by a comparison with theidentity 1. From our forgoing assumptions on G , this can be done in time ˜ O ( n ). Finally, as | S | ∈ O (log n ), generators of B can be obtained from ˜ O ( | G : B | ) group products and membershiptests, using CayleyGraph ( S, B ); all this can be done in time ˜ O ( | G : B | n ) ⊂ ˜ O ( n . ) since | G : B | ∈ (log n ) O ((log log n ) ) by Theorem 2.5. (cid:3) We now introduce a tool that lets us find a complement to a normal Hall π big n -subgroup. BigSplit
Given a group G of order n ∈ Υ, Return a subgroup H G such that G = H ⋉ B , where B is the Hall π big n -subgroup.To solve BigSplit we need a brief detour into a generic model for encoding groups via presen-tations, cf. [27, Section 1.4]: The free group F [ X ] on a given alphabet X is formed by creating adisjoint copy X − of the alphabet and treating the elements of F [ X ] as words over the disjoint union X ⊔ X − , including the empty word 1. The latter serves as the identity and word concatenationis the group product; to impose the existence of inverses we apply rewriting rules xx − → x − x → x ∈ X and corresponding x − ∈ X − . For a group G , tuple g ∈ G X , and word w ∈ F [ X ], we assign an element w ( g ) ∈ G by replacing each variable x ± in w with the value g x ∈ G and g − x ∈ G , respectively, and then evaluating the corresponding product in G . The map-ping w w ( g ) is a homomorphism ˆ g : F [ X ] → G , whose kernel ker ˆ g = { w ∈ F [ X ] : w ( g ) = 1 } isa normal subgroup of F [ X ]. If G is generated by the image S = { g x : x ∈ X } and R generates ker ˆ g as a normal subgroup, then the pair h S | R i is a presentation of G , where R is a set of relations for G relative to S . Note that h S | R i carries all the information necessary to describe G up toisomorphism; however, in such an encoding isomorphism testing may be even become undecidable,see [27, Section 1.9].Our interest in presentations is to produce a relatively small number of equations whose solutionshelp to solve BigSplit ; for that purpose the following will suffice.
Proposition 3.3.
Let G be a group of order n ∈ Υ with Hall π big n -subgroup B . There is an ˜ O ( n . ) -time algorithm to compute a presentation h S | R i of the quotient G/B such that | S | ∈ O (log n ) and | R | ∈ (log n ) O ((log log n ) ) , and each w ∈ R is a word in S of length O (log n ) .Proof. As shown above, we find G = h S i with | S | ∈ O (log n ). Use CayleyGraph and Corollary 3.2to get a transversal T for B in G , generators for B , and a rewriting algorithm that given g ∈ G ,finds a word ¯ g in S with ¯ gB = gB . Choose a set X = { x g : g ∈ S } of variables, and for each g ∈ G define w g ∈ F [ X ] as the word in X ∪ X − produced by replacing each u ∈ S in g with x u . Now R = { w t x s w − ts : t ∈ T , s ∈ S } is a set of relations for G/B relative to { sB : s ∈ S } , cf. [26, p. 112].Note that | R | | G : B | · | S | , and | G : B | ∈ (log n ) O ((log log n ) ) by Theorem 2.5. Also computing w g is dominated by the time ˜ O ( | G : B | ) it takes to compute ¯ g . We do this on O ( | S | · | G : B | )elements for a total time of (log n ) O ((log log n ) ) ∈ O ( n . ). Our relators have length O ( | G : B | ),but there is a ˜ O ( n )-time algorithm to replace such relators with ones of length O (log n ), see [26,Lemma 4.4.2]. (cid:3) Remark 3.1.
Babai-Luks-Seress, Kantor-Luks-Marks and others (see the bibliography in [14] and[26, Section 6]) developed various algorithms to construct presentations of (quotient) groups ofpermutations. Their complexities range from polynomial-time, to polylogarithmic-parallel (NC),to Monte-Carlo nearly-linear time, and they produce presentations that can be considerably smallerthan what we obtain in Proposition 3.3. Though it is not necessary for our complexity goals, weexpect that a better analysis and a better performing implementation would use such methodsinstead of our above brute-force approach.
GROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS
Proposition 3.4.
BigSplit is in time ˜ O ( n . ) .Proof. BigSplit is solved via the function
Complement discussed in [14, Section 3.3]; we brieflysketch the approach. Let G = h S i with S = { s , . . . , s d } and d ∈ O (log n ). Use the algorithmof Proposition 3.3 to get a presentation h x , . . . , x d | R i for G/B , such that each x i = x s i asdefined in the proof of Proposition 3.3. Every complement H to B , if it exists, is generated by { s m , . . . , s d m d } for some m , . . . , m d ∈ B , and such a generating set satisfies the relations in R ,cf. [24, (2.2.1)]. We attempt to compute m , . . . , m d by solving the system of equations resultingfrom w ( s m , . . . , s d m d ) = 1 with w running over R : recall that w ( s m , . . . , s d m d ) ∈ G is definedas w ( g ) with g = ( s m , . . . , s d m d ), see the remarks above Proposition 3.3. A complement to B exists if and only if this equation system has a solution. Since each w ( s , . . . , s d ) lies in thefinite cyclic group B , this system can be described by an integral matrix with log n variables and(log n ) O ((log log n ) ) equations; using the algorithms of [28], it can be solved via Hermite NormalForms in time (log n ) O ((log log n ) ) . (cid:3) Proofs of the main results
Proof of Theorem 1.1: isomorphism testing.
Proof of Theorem 1.1.
Given two binary maps [ n ] × [ n ] → [ n ], we decide that n ∈ Υ in time O ( n ),and we use Theorem 1.2 to decide whether these maps describe Cayley tables. If so, we have beengiven two groups G and ˜ G of order n ∈ Υ, and we can use Theorem 2.6 to find generators forsubgroups
H, B G and ˜ H, ˜ B ˜ G with G = H ⋉ B and ˜ G = ˜ H ⋉ ˜ B . Having generators of thesesubgroups, we can define homomorphisms θ and ˜ θ such that G = H ⋉ θ B and ˜ G = ˜ H ⋉ ˜ θ ˜ B . Since n ∈ Υ, we know that B and ˜ B are cyclic, hence B and ˜ B are isomorphic if and only if | B | = | ˜ B | .Moreover, since | H | , | ˜ H | (log n ) O ((log log n ) ) we can test isomorphism H ∼ = ˜ H using brute-forcemethods in time (log n ) O ((log log n ) ) . If H ∼ = ˜ H and B ∼ = ˜ B is established, then we can identify H = ˜ H and B = ˜ B , and test G ∼ = ˜ G by using Lemma 2.1: since B is cyclic, Aut( B ) is abelian, andso Condition (1) reduces to ˜ θ ( α ( h )) = θ ( h ) for all h ∈ H , which we can test by enumerating Aut( H )and looking for a suitable α ; since | H | is small, such a brute-force enumeration is efficient. (cid:3) Proof of Theorem 1.2: recognizing groups.
Similar to [11, Chapter 2], our strategy forrecognizing groups uses Cayley’s Theorem [24, (1.6.8)]. The latter implies that the rows of an n × n group table can be interpreted as permutations which form a regular permutation group on [ n ],that is, the group is transitive on [ n ] and has trivial point-stabilizers, see [26, Section 1.2.2].A new idea is to exploit that groups of order n can be specified by generating sets of size log n ,so some log n rows determine the entire table. Once the input is verified to be a latin square,our approach is to define a permutation group generated by O (log n ) rows, and then compare itsCayley table with the original table. In more abstract terms, our algorithm creates an instance of anabstract permutation group data type, as defined in [26, Section 3]. That data type is guaranteedto be a group and so the promise is converted into a computable type-check: We confirm that thegroup we create in this new data type is the one specified by the original table; the proof givenbelow makes this argument specific. This methodology of removing a promise by appealing to atype-checker generalizes; we refer to our forthcoming work [9] for more details. Proof of Theorem 1.2.
Let ∗ : [ n ] × [ n ] → [ n ] be the multiplication defined by the table T . In time˜ O ( n ) we verify that T is a latin square (if not, return false) and arrange that T is reduced , that is,its first row and column have 1 , . . . , n in order. Now T describes a loop L = ([ n ] , ∗ ) with identity 1.For i ∈ [ n ] denote by λ i ∈ Sym n the map [ n ] → [ n ] defined by left multiplication λ i ( a ) = i ∗ a .Since T is reduced, λ i is given by the i -th row of T . Note that Λ = h λ i : i ∈ [ n ] i is a subgroupof Sym n , and L is a group if and only if Λ is a regular permutation group on [ n ], if and only if λ i λ j = λ i ∗ j for all i, j ∈ [ n ], see [11, Theorems 2.16 & 2.17]. Since T is reduced, it follows that Λ ROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS 9 is a transitive subgroup of Sym n . Thus, Λ is regular if and only if the stabiliser Λ of 1 ∈ [ n ] inΛ is trivial. We now show how to find a generating set of Λ of size O (log n ) and prove that Λ isregular, or prove that L is not a group and return false.For a subset S ⊆ [ n ] define Λ( S ) = h λ i : i ∈ S i . Since T is reduced, λ i (1) = i for every i ∈ [ n ].Let Λ( S )(1) = { λ (1) : λ ∈ Λ( S ) } and Λ( S ) = { λ ∈ Λ( S ) : λ (1) = 1 } be the orbit and stabiliserof 1 in Λ( S ), respectively. Note that for each S ⊆ [ n ] we have Λ( S ) Λ. We describe how to usethe orbit-stabiliser algorithm [26, Section 4] to get Λ( S )(1), a generating set for Λ( S ) (so-calledSchreier generators), and for each i ∈ Λ( S )(1) an transversal element λ ( i ) ∈ Λ( S ) with λ ( i ) (1) = i .We start by choosing a subset S ⊆ [ n ] of size O (log n ) and by copying { λ i : i ∈ S } to aseparate tape; we can assume that 1 ∈ S . In the following we will mainly work with this shorttape; scanning it takes time ˜ O ( n ). We will store and represent an element λ i k · · · λ i ∈ Λ( S ) as asequence ( i k , . . . , i ) with each i j ∈ S . Initialize T = { [1 , λ ] } and R = {} ; the set T will store data[ x ; i k , . . . , i ], meaning that λ ( x ) = λ i k · · · λ i is the chosen transversal element that maps 1 to x .The set R will store pairs [ i k , . . . , i ; j m , . . . , j ] encoding the relation λ i k · · · λ i (1) = λ j m · · · λ j (1).The construction of the set of relations R will show that we always have k >
2, and λ j m · · · λ j and λ i k − · · · λ i are both transversal elements.We now describe an iteration that will successfully build T and R ; it will turn out that |T | n and | R | ∈ ˜ O ( n ). Take the first element of T , say [ x ; i k , . . . , i ] and then run over each generator λ u with u ∈ S . In time ˜ O ( n ), we evaluate y = λ u ( x ) and look for an entry in T with left-handside y . If it exists, say [ y ; j m , . . . , j ] ∈ T , then we know that λ j m · · · λ j (1) = y = λ u λ i k · · · λ i (1),and we add [ u, i k , . . . , i ; j m , . . . , j ] to R . If it does not exist, then we add [ y ; u, i k , . . . , i ] to T ,and so λ ( y ) = λ u λ i k · · · λ i is a new transversal element. We repeat this process until we have notincreased the size of T anymore; this requires ˜ O ( n ) iterations, see [26, Section 4]. Thus, as long asall word lengths k, m are ˜ O ( n ), the work described in this paragraph takes time ˜ O ( n ).Once the work of the previous paragraph is finished, we have Λ( S )(1) = { i : [ i, ∗ ] ∈ T } , and wetest whether Λ( S ) is transitive by checking whether |T | = n . If not, then choose k ∈ [ n ] \ Λ( S )(1),replace S by S ∪ { k } , and repeat the orbit-stabiliser algorithm of the previous paragraph. Byconstruction, λ k (1) = k , so the new orbit and hence the new group Λ( S ) have increased in size. Wecan increase | Λ( S ) | only O (log n ) times, so we repeat the work of the previous paragraph O (log n )times. In conclusion, in time ˜ O ( n ) we can achieve that Λ( S ) is transitive on [ n ]. (Note that theoriginal loop L = ([ n ] , ∗ ) is transitive, so we will always achieve this.)We have now obtained a Schreier tree, a generating set of size Λ( S ), and a transversal; we usethe algorithm of [26, Lemma 4.4.2] in time ˜ O ( n ) to achieve a shallow Schreier tree, meaning thatevery [ x ; i k , . . . , i ] ∈ T has k bounded by O (log n ), that is, every λ ( x ) is a word of length O (log n )in the generators.It remains to check that the stabiliser Λ( S ) is trivial. By the orbit-stabiliser algorithm, thisstabiliser is generated by the Schreier generators encoded by the relations stored in R . It followsthat Λ( S ) is trivial if and only if for each of the ˜ O ( n ) relations [ i k , . . . , i ; j m , . . . , j ] in R wehave λ i k · · · λ i = λ j m · · · λ j . Note that by construction m, k ∈ O (log n ) and we can multiply twogenerators in time ˜ O ( n ), thus, in time ˜ O ( n ) we can check equality. In total, checking that Λ( S ) istrivial can be done in time ˜ O ( n ). In conclusion, we have found a regular permutation group Λ( S )contained in Λ.It remains to show that Λ Λ( S ). For this it is sufficient to show that each λ i ∈ Λ( S ); notethat the latter holds if and only if λ i = λ ( i ) is a transversal element. For this we first sort T so thatits elements correspond to λ (1) , . . . , λ ( n ) ; note that T has n entries of length O ((log n ) ) bits, sowe can sort T in time ˜ O ( n ). Once sorted, we rewrite every entry [ x ; i k , . . . , i ] as the permutation λ ( x ) ; this requires to multiply λ i k · · · λ i ; this can be done for all x in time ˜ O ( n ). Lastly, in time˜ O ( n ) we scan the original table T and compare each λ x = λ ( x ) . (cid:3) Remark 4.1.
Having converted our Cayley table into a regular permutation group G , we nowhave a generating set S of size O (log n ) and a corresponding Schreier tree; both are stored in aseparate tape of length ˜ O ( n ). This is the tape that is used whenever we operate in the group; theoriginal input tapes will never be revisited. As mentioned in the proof of Theorem 1.2, we mayassume that the Schreier trees are shallow , that is, they have depth bounded by O (log n ), see [26,Lemma 4.4.2]. The algorithms we now use are as follows. Each g ∈ G is a node in the Schreiertree and there is a unique path from the origin to g ; if g , . . . , g k ∈ S are the labels of that path,then g = g k · · · g ; note that k ∈ O (log n ) since the Schreier tree is shallow. We now describehow to compute the product of these labels. Recall that g , . . . , g k are permutations on [ n ], so wecan compute the image of 1 ∈ [ n ] under g k · · · g , by looking up i = g (1), i = g ( i ), etc, untilwe obtain u = g k · · · g (1). This scan occurs on the short tape in time ˜ O ( n ). Since the group isregular, there is a unique g ∈ G with u = g (1), which determines g = g k · · · g . To multiply elements g k · · · g and g ′ j · · · g ′ , we merely concatenate the generators, and continue with this word, yieldinga (log n ) O (1) -time multiplication. We note that none of our product lengths exceed (log n ) O (1) .To compare g k · · · g and g ′ j · · · g ′ with k, j ∈ O (log n ), we determine and compare g k · · · g (1) and g ′ j · · · g ′ (1) in time ˜ O ( n ). More details of these methods are given in [26, p. 85–86].5. Conclusion and outlook
We have shown that when restricted to a dense set Υ of group orders, testing isomorphism of groupsof order n ∈ Υ given by Cayley tables can be done in time ˜ O ( n ); this significantly improves theknown general bound of n O (log n ) . The set Υ includes all square-free integers ( ≈
60% of all orders),and we note that | Υ ∩ { , , . . . , k }| / k is already approximately 0.535, 0.618, and 0.702 for k = 8 , ,
10, respectively; moreover, one can decide if n ∈ Υ in time O ( n ).We have proved that groups of these orders admit a computable factorisation G = H ⋉ B withthe following useful property: firstly, the H ard group theory of G is captured in H , but | H | is small compared to | G | so brute-force methods can be applied to H ; secondly, the B ig number theory of | G | is captured by | B | , but B is cyclic, hence its group theory is easy . These decompositions existfor a dense set of group orders, so we expect this will be useful for other computational tasks aswell. In fact, we will exploit properties of these decompositions in future work: This paper is partof our program to enhance group isomorphism, see [7, 8] for recent work, and we plan to extendthe present results to other input models. Specifically, in our current work [9] we develop a newblack-box input model for groups (based on Type Theory) that does not need a promise that theinput really encodes a group, so algorithms for this model can be implemented within the usualpolynomial-time hierarchy. Due to Theorem 1.2, the algorithms presented here do also not requirea promise that the input tables describe groups. We conclude by mentioning that our algorithmfor isomorphism testing can be adapted to find a single isomorphism, generators for the set of allisomorphisms, or to prescribe a canonical representative of the isomorphism type of a single group. Acknowledgments
Both authors thank Joshua Grochow and Youming Qiao for comments on earlier versions of thisdraft. They also thank the Newton Institute (Cambridge, UK) where some of this research tookplace and we recognize the funding of EPSRC Grant Number EP/R014604/1. Dietrich was sup-ported by an Australian Research Council grant, identifier DP190100317. Wilson was supportedby a Simons Foundation Grant, identifier
ROUP ISOMORPHISM IS NEARLY-LINEAR TIME FOR MOST ORDERS 11
Appendix A. Proofs of Theorems 2.4–2.6
A.1.
Number theory: Proof of Theorem 2.4.
Proof of Theorem 2.4.
Erd˝os-P´alfy [10, Lemma 3.5] showed that almost every n ∈ N has the prop-erty that if a prime p > log log n divides n , then p ∤ n ; thus, n = p e . . . p e k k b with b square-free,every prime divisor of b is greater than log log n , and p , . . . , p k log log n are distinct primes. Let x > N ( x ) of integers 0 < n x which are divisible by a prime p log log n such that the largest p -power p e dividing n satisfies p e > log n . We want to show that N ( x ) /x → x → ∞ ; this proves that for almost all integers n , if p e | n with p log log n , then p e log n . To get an upper bound for N ( x ), we considerintegers between √ x and x with respect to the above property, and add √ x for all integers between1 and √ x . Note that if p e > log n , then e > log log n/ log p . Since we only consider √ x n x ,this yields e > c ( x ) where c ( x ) = log log √ x/ log log log x . Note that c ( x ) → ∞ if x → ∞ , thus N ( x ) √ x + X ⌊ log log √ x ⌋ k =2 xk c ( x ) √ x + x Z log log √ x y c ( x ) d y = √ x + x − c ( x ) (cid:20) √ x ) c ( x ) − − c ( x ) − (cid:21) . Since 1 / (1 − c ( x )) → N ( x ) √ x + x (cid:12)(cid:12)(cid:12)(cid:12) − c ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) − √ x ) c ( x ) − + 12 c ( x ) − (cid:21) √ x + x (cid:12)(cid:12)(cid:12)(cid:12) − c ( x ) (cid:12)(cid:12)(cid:12)(cid:12) (cid:20) c ( x ) − (cid:21) ;thus N ( x ) = o ( x ), since N ( x ) /x √ x/x + (cid:12)(cid:12) / (1 − c ( x ))2 c ( x ) − (cid:12)(cid:12) → x → ∞ . This provesthat the set Υ of all positive integers satisfying conditions a,c) in Definition 2.3 is dense. By [10,Lemmas 3.5 & 3.6], the set Υ of positive integers n satisfying conditions b,c) is dense as well. Aninclusion-exclusion argument proves that Υ = Υ ∩ Υ is dense. (cid:3) A.2.
Splitting theorems: Proofs of Theorems 2.5 & 2.6.
Proof of Theorem 2.5.
Let G be a group of order n ∈ Υ; we first show that G has a normal Hall π big n -subgroup; in fact, we show that if G is solvable, then there is a normal Hall π n -subgroup.First, suppose that G is solvable. We show that G has a normal Sylow p -subgroup for every p ∈ π n . Let q = p be a prime dividing n , and let A be a Hall { p, q } -subgroup of G of order p e q f ; see[24, Section 9.1] The Sylow Theorem [24, (1.6.16)] shows that the number h p of Sylow p -subgroupsof A divides q f (and hence n ) and p | ( h p − p | n is isolated, we have h p = 1 and A hasa normal Sylow p -subgroup. Now fix a Sylow basis P = { P , . . . , P s } for G , that is, a set of Sylowsubgroups, one for each prime dividing n , such that P i P j = P j P i for all i and j ; see [24, Section 9.2].Let P = P u be the Sylow p -subgroup for G in P . Since G = P · · · P s , every g ∈ G can be writtenas g = g . . . g s with each g j ∈ P j . Since P P j = P j P , the group P P j is a Hall { p, p j } -subgroup. Asshown above, P E P P j , so each g j P = P g j . Thus, gP = g . . . g s P = P g . . . g s = P g , so P E G .Second, suppose that G is non-solvable. We show that G has a normal Sylow p -subgroup forevery p ∈ π big n . Being non-solvable, G has a non-abelian simple composition factor, so | G | is divisibleby 4, see [5, p. 155]. Since n ∈ Υ, this implies that 2 ν ( n ) log n , so ν ( n ) log log n < p forevery p ∈ π big n . We claim that p ∤ | G : O ∞ ( G ) | where O ∞ ( G ) E G is the largest normal solvablesubgroup. The socle of G/O ∞ ( G ) decomposes as T × · · · × T ℓ where each T i is non-abelian simpleand a minimal normal subgroup of G/O ∞ ( G ), see [26, pp. 157–159]. Note that p ∤ | T i | for each i since p ∈ π big n . By the Classification of Finite Simple Groups, a prime r divides | Aut( T i ) | only if r divides | T i | , as seen from the list of known orders of simple groups and their outer automorphism The finite simple groups (Classification Theorem of Finite Simple Groups) are listed in [30, Section 1.2]; the ordersof these groups and the orders of their automorphism groups are described in various places in said book. Simply groups. Let PKer( G ) /O ∞ ( G ) be the kernel of the permutation representation G/O ∞ ( G ) → Sym ℓ induced by the conjugation action on { T , . . . , T ℓ } ; this kernel embeds into Aut( T ) × . . . × Aut( T ℓ ).Assume, for a contradiction, that p divides | G : O ∞ ( G ) | . By assumption, p ∤ | Aut( T i ) | for each i ,which forces p | | G : PKer( G ) | and p ℓ . Every T i has even order by the Odd-Order Theorem,see [30, p. 2], so 2 ℓ | n and ν ( n ) > ℓ ; now p ℓ contradicts p > ν ( n ), which we have shownabove. This forces p ∤ | G : O ∞ ( G ) | , so the Sylow p -subgroup P lies in O ∞ . Since p is also isolatedin | O ∞ ( G ) | , the proof of the solvable case shows P E O ∞ ( G ). Since O ∞ ( G ) is characteristic in G ,we know that P is normal in G .In conclusion, for every p ∈ π big n there is a normal Sylow p -subgroup G p in G . Since n ∈ Υ and p > log log n , this subgroup is cyclic of order p . If p, q ∈ π big n are distinct, then G p , G q E G implies G q G p ∼ = G p × G q ∼ = C pq , a cyclic group of order pq . It follows that G has a normal cyclic Hall π big n -subgroup B , and G = H ⋉ B for some H G by the Schur-Zassenhaus Theorem [24, (9.1.2)].It remains to prove that | H | (log n ) O ((log log n ) ) . Recall from Definition 2.3 that n = ab suchthat a (log n ) log log n and if p ∈ π big n , then p | b and p ∤ n . In particular, | H | = ab/z , where z is the product of the primes in π big n . It remains to show that b/z (log n ) O ((log log n ) ) . If p is a prime divisor of b/z , then p > log log n and p / ∈ π big n , that is, p ∤ n and p | n is isolated,but not strongly isolated. Thus, there is some non-abelian simple group T of order dividing n with p | | T | . In the remainder of this proof we show that the non-abelian composition factorsof H all together contribute at most (log n ) O ((log log n ) ) to the order of H ; this will imply that b/z (log n ) O ((log log n ) ) , which then completes the proof.We first show that the number f of non-abelian composition factors of H satisfies f log log n .Distinct simple groups intersect trivially and every non-abelian simple group has order divisibleby 4, see [5, p. 155], so 4 f | n . Now n ∈ Υ forces 4 f log n , hence f log log n . Thus, the proofis complete if every non-abelian composition factor of H has order at most (log n ) O ((log log n )) .One can see from the known factorized orders of the finite non-alternating simple groups thatevery such group T has a distinguished prime power divisor r m | | T | with m > | T | ( r m ) O ( m ) :This is trivially true for the 26 sporadic groups; for the other non-alternating simple groups thisfollows because they are representable as quotients of groups of d × d matrices over a field of order r e , and then m = de . Now if the order of T divides n ∈ Υ, then m > r log log n and r m log n , hence m log log n , and | T | (log n ) O ((log log n )) , as claimed.If T ∼ = Alt k is alternating of order k ! / k k = 2 k log k , then the distinguished prime power divisoris 2 ν ( k !) − . Legendre’s formula [21, Theorem 2.6.4] shows that ν ( k !) = k − s ( k ) where s ( k ) log k is the number of 1’s in the 2-adic representation of k . Since n ∈ Υ, we have 2 ν ( k !) − log n , so k − log( k ) − ν ( k !) − log log n . This shows that 2 k k log n , and so | T | (2 k log n ) log k .Note that | T | = k ! / n , and so Stirling’s formula ln( k !) = k ln k − k + O (ln k ) shows that k log 2 n for large enough k . This yields | T | (log n ) O (log log n ) , as claimed. (cid:3) Proof of Theorem 2.6.
Let G be a group of order n ∈ Υ. By Theorem 2.5 and Corollary 3.2, wecan construct generators and a membership test for the cyclic Hall π big n -subgroup B of G . Withthis we can use Proposition 3.3 to construct a complement H , thus G = H ⋉ B . The complexitystatement follows from the results we have used. (cid:3) References [1] L. Babai. Graph isomorphism in quasi-polynomial time. arXiv:1512.03547.[2] L. Babai, P. Codenotti, Y. Qiao. Polynomial-time isomorphism test for groups with no abelian normal subgroups.LNCS - Automata, Languages, and Programming - Proceedings, ICALP, Springer, Warwick, UK, (2012) 51-62.[3] L. Babai, Y. Qiao. Polynomial-time isomorphism test for groups with abelian Sylow towers. STACS, DagstuhlPublishing, Paris, (2012) 453-465.for the convenience of the reader, we refer to en.wikipedia.org/wiki/List of finite simple groups fora concise list of these orders.
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