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Code Equivalence and Group Isomorphism Code Equivalence and Group Isomorphism L´aszl´oBabai,∗ Paolo Codenotti, Joshua A. Grochow flaci, paoloc, [email protected] (University of Chicago) and Youming Qiaoy [email protected] (Tsinghua University) Abstract ited for pointing out that if one of the groups is gener- The isomorphism problem for groups given by their ated by k elements then isomorphism can be decided in k+O(1) multiplication tables has long been known to be solv- time n where n is the order of the groups; indeed able in time nlog n+O(1). The decades-old quest for a one can list all isomorphisms within this time bound (cf. polynomial-time algorithm has focused on the very dif- [27]). Since k ≤ log n for all groups, this in particular log n+O(1) ficult case of class-2 nilpotent groups (groups whose quo- gives an n -time algorithm for all groups (log tient by their center is abelian), with little success. In to the base 2) and a polynomial-time algorithm for fi- this paper we consider the opposite end of the spec- nite simple groups (because the latter are generated by trum and initiate a more hopeful program to find a 2 elements, a consequence of their classification [14]). polynomial-time algorithm for semisimple groups, de- In spite of considerable attention to the problem fined as groups without abelian normal subgroups. First over the past quarter century, no general bound with a we prove that the isomorphism problem for this class sublogarithmic exponent has been obtained. can be solved in time nO(log log n). We then identify cer- While the abelian case is easy (O(n) according to 2 tain bottlenecks to polynomial-time solvability and give Kavitha [19], improving Savage's O(n ) [30] and Vikas's a polynomial-time solution to a rich subclass, namely O(n log n) [34]), just one step away from the abelian the semisimple groups where each minimal normal sub- case lurk what appear to be the most notorious cases: group has a bounded number of simple factors. We nilpotent groups of class 2. These groups G are defined relate the results to the filtration of groups introduced by the property that the quotient G=Z(G) is abelian, by Babai and Beals (1999). where Z(G) is the center of G. No complete structure One of our tools is an algorithm for equivalence theory of such groups is known; recent work in this of (not necessarily linear) codes in simply-exponential direction by James Wilson [35, 36] commands attention. time in the length of the code, obtained by modifying Recently, other special classes of solvable groups Luks's algorithm for hypergraph isomorphism in simply- have been considered; the isomorphism problem of exponential time in the number of vertices (FOCS extensions of an abelian group by a cyclic group of 1999). relatively prime order has been solved very efficiently We comment on the complexity of the closely re- (sublinear time in the black-box model) [22]. We note lated problem of permutational isomorphism of permu- that the structure of such groups is well understood. tation groups. While class-2 nilpotent groups have long been recog- nized as the chief bottleneck in the group isomorphism 1 Introduction problem, this intuition has never been formalized. The ultimate formalization would reduce the general case to 1.1 Group isomorphism - bottlenecks and ap- this case. As a first step, we consider a significant class proach. The isomorphism problem for groups asks to without a chance of a complete structure theory at the determine if two groups, given by their Cayley tables opposite end of the spectrum: groups without abelian (multiplication tables), are isomorphic. Tarjan is cred- normal subgroups. Following [29], we call such groups semisimple1. Our project is to show that semisimple ∗L´aszl´oBabai's work was supported in part by NSF Grant CCF-0830370. groups admit a polynomial-time isomorphism test. yYouming Qiao's work was supported in part by the Na- tional Natural Science Foundation of China Grant No.60553001, 1 and the National Basic Research Program of China Grant We note that authors use the term `semisimple group' in Nos.2007CB807900, 2007CB807901. several different meanings (see e. g. [33]). 1.2 A general result. The solvable radical Rad(G) Theorem 1.2. Isomorphism of semisimple groups G of a group G is the unique maximal solvable normal and H of order n can be decided, and the coset of subgroup of G. A group G is semisimple if and only isomorphisms found, in time nO(1)+c log(t(G))), where if Rad(G) = 1. For every group G, the quotient c = 6= log(60) ≈ 1:0158. G= Rad(G) is semisimple. This fact indicates the rich- ness of the class of semisimple groups. We prove this result in Section 6 (see Corollary 6.2). Our first result, to be proved in Section 4 (see Note that t(G) ≤ log60 n, and hence this result sub- Corollary 4.2), concerns the entire class. sumes Theorem 1.1 (but the algorithm of Theorem 1.1 is much simpler). Theorem 1.1. Isomorphism of two semisimple groups Every semisimple group is an extension of a group of order n can be decided in time nO(1)+c log log n, where G with t(G) = 1 by a permutation group of logarithmic c = 1= log(60) ≈ 0:16929. In fact, all isomorphisms can degree (Fact 7.3). Therefore a key ingredient of the be listed within this time bound. yet unsolved part of the project will be to decide permutational isomorphism of permutation groups of Remark. Because the algorithm above lists all the degree k in time polynomial in 2k and the order of the isomorphisms, we cannot hope to get a better bound groups. That doing so is indeed necessary is shown on the running time for pairs of groups with that in Prop. 7.1. While we cannot claim that it is also many isomorphisms. Such groups do indeed exist. For k sufficient (cf. Appendix Section 7.6), we believe that a example, consider the group G = A5 , the direct product solution of the stated complexity for the permutational of k copies of the alternating group of order 60. The k k c log log n isomorphism problem, combined with the methods of group A5 is semisimple and has 120 k! > n k the present paper, will get us close to a polynomial- automorphisms (n = jGj = 60 ), where c = 1= log(60). time solution of group isomorphism for all semisimple Recall that the trivial algorithm to check isomor- groups. We solve the case of bounded orbits in the phism takes time nO(1)+k, where k is the number of required time (see Theorem 7.2). We note that this case generators of our groups2. We point out that Theo- includes equivalence of linear codes over prime fields of rem 1.1 is not a special case of the nO(1)+k bound. bounded order (see Proposition 7.2) Fact 1.1. There exist semisimple groups which require 1.4 Codes. We reduce the isomorphism problem for at least log120 n generators. semisimple groups to equivalence of group codes. We For example, Sk is semisimple (where S is the sym- consider the code equivalence problem as a separate 5 5 problem of interest in its own right. A code of length n metric group of degree 5 and order 120), but every set A k k over a finite alphabet Γ is a subset of Γ for some set of generators of S5 has size at least k, since S5 has a A quotient isomorphic to k. Here k = log (n). A with jAj = n. An equivalence of the codes A ⊆ Γ Z2 120 and B ⊆ ΓB is a bijection A ! B that takes A to B. If 1.3 The main result. We now deal with cases when jΓj = 2 then the code is a Boolean function or hyper- it is not possible to list all the isomorphisms within graph, so the code equivalence problem is a generaliza- tion of the hypergraph isomorphism problem. Modify- the desired time bound. The set of isomorphisms n of two groups G and H is either empty or a coset ing and extending Luks's C dynamic programming al- Aut(G)σ of Aut(G), which we will represent by a list gorithm for hypergraph isomorphism [26] to treat code of generators of the automorphism group of G and a equivalence, we obtain the following result, proved in particular isomorphism σ : G ! H. Section 5.2. Every minimal normal subgroup is characteristi- Theorem 1.3. The set of equivalences of two codes of cally simple, and hence it is the direct product of iso- length n over an alphabet of size k can be found in time morphic simple groups. (See Section 2.5 for definitions.) (ck)2n, for some absolute constant c. We parametrize our groups G by a parameter t(G) and solve the case of bounded t(G) in polynomial time, As before, the set of equivalences is a coset, given O(log(t(G)+1)) and the general case in time n . We define by generators and a coset representative. t(G) as the smallest t such that each minimal normal We remark that our algorithm, while inspired by subgroup of G has at most t simple factors. Our main Luks's, is different from his even in the special case result is the following. of hypergraph isomorphism. We obtain some simpli- fication by eliminating a divide-and-conquer aspect of 2Throughout this paper, n denotes the order of the groups to Luks's algorithm; the cost is somewhat lesser efficiency.
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