## Computing in quotient groups (1990)

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Venue: | Proceedings of the 22nd ACM Symposium on Theory of Computing |

Citations: | 17 - 6 self |

### BibTeX

@INPROCEEDINGS{Kantor90computingin,

author = {William M. Kantor and Eugene M. Luks},

title = {Computing in quotient groups},

booktitle = {Proceedings of the 22nd ACM Symposium on Theory of Computing},

year = {1990},

pages = {524--563},

publisher = {ACM Press}

}

### Years of Citing Articles

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### Abstract

We present polynomial-time algorithms for computation in quotient groups G/K of a permutation group G. In effect, these solve, for quotient groups, the problems that are known to be in polynomial-time for permutation groups. Since it is not computationally feasible to represent G/K itself as a permutation group, the methodology for the quotient-group versions of such problems frequently differ markedly from the procedures that have been observed for the K = 1 subcases. Whereas the algorithms for permutation groups may have rested on elementary notions, procedures underlying the extension to quotient groups often utilize deep knowledge of the structure of the group. In some instances, we present algorithms for problems that were not previously known to be in polynomial time, even for permutation groups themselves (K = 1). These problems apparently required access to quotients. 1.

### Citations

309 |
Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes
- Babai, Moran
- 1988
(Show Context)
Citation Context ...that between finding graph automorphism-groups and ISO. The analogy has been reinforced by Babai and Moran, who show that the NP-completeness of COSANTERS would imply the same collapse ~ = IIp -- AM (=-=[BM]-=-). We remark, finally, that problems such as I-VI are not considered difficult in practical computation, and systems such as CAYLEY [Ca] allow quite efficient implementations. This should be no surpri... |

307 |
The Theory of Groups
- HALL
- 1959
(Show Context)
Citation Context ...S in G is the smallest normal subgroup of G containing S, namely (SG). IfH < G then the core of H in G, Coree(H), is the largest normal subgroup of G contained in H, namely N{H a ]g 6 G}. We refer to =-=[Ha]-=- for a discussion of Sylow's Theorem: (i) If p is a prime then a Sylow p-subgroup of G is a p-subgroup whose order is the p-part of IGI; (ii) any psubgroup of G is contained in a Sylow p-subgroup; (ii... |

218 |
Finite permutation groups
- Wielandt
- 1964
(Show Context)
Citation Context ...re isomorphic to a subgroup of Sym(d); in particular, Fd contains all solvable groups. A most significant effect of this restriction on a class of groups is that the primitive permutation groups (see =-=[Wi]-=-) in the class have polynomially-bounded order [BCP]. (Primitive groups arise naturally as the base cases in certain divide-andconquer procedures; see, e.g., [Lul]). There are fairly elementary proced... |

160 |
Proofs That Yield Nothing but Their Validity and a Methodology of Cryptographic Protocol Design
- Goldreich, Micali, et al.
- 1991
(Show Context)
Citation Context ...s, isomorphism is known to be testable in linear time [BK], [Ku]. Furthermore, there is strong evidence that ISO is not NPcomplete, else the polynomial-time hierarchy would collapse to E~ = H E = AM (=-=[GMW]-=-). Nevertheless, ISO has stubbornly resisted attempts to place it in polynomialtime. (At present the best algorithm for general graphs has worst-case complexity exp(c~) [BELl.) Consider now the follow... |

141 |
Isomorphism of graphs of bounded valence can be tested in polynomial time
- Luks
- 1982
(Show Context)
Citation Context ...the primitive permutation groups (see [Wi]) in the class have polynomially-bounded order [BCP]. (Primitive groups arise naturally as the base cases in certain divide-andconquer procedures; see, e.g., =-=[Lul]-=-). There are fairly elementary procedures for testing membership in Fd (see [Lul, §4]). For our purposes, it is essential only that d be fixed; the specific value of d would play a role in more precis... |

75 |
Canonical labeling of graphs
- Babai, Luks
- 1983
(Show Context)
Citation Context ... has developed for computing with permutation groups. A major stimulus for this activity was the application to the graph isomorphism problem (ISO), for early work ([Bal], [FHL], [Lull, [Mill, [Mi2], =-=[BL]-=-) used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ([BKL], [Lu2], [Ne], [KT], [Kal], ... |

70 |
On the complexity of matrix group problems I
- Babai, Szemerédi
- 1984
(Show Context)
Citation Context ... for some listed set A (r(M) being specified on generators of A, in which case we might also need to verify that ~r(M) C Aut(A)). More generally, we may suppose H is a black box group in the sense of =-=[BS]-=-. When K = 1, P3(i) is contained in [FHL]; the analogue for quotient groups is immediate since ((H/K) a/K) = (Ha)/K. P3(ii) is an easy consequence of the observation that H is subnormal in G iff H is ... |

38 |
Group-Theoretic Algorithms and Graph Isomorphism, volume 136 of LNCS
- Hoffmann
- 1982
(Show Context)
Citation Context ...since ((H/K) a/K) = (Ha)/K. P3(ii) is an easy consequence of the observation that H is subnormal in G iff H is subnormal in (HG). 527When K = 1, P4(i) is an easy application of results in [FHL] (see =-=[Ho]-=-, or [CFL] where it is directly reduced to 3.2); the general case is an immediate consequence. In view ofP3(ii), P4(ii) follows at once. For g = 1, P4(iii) is in [Lul, §4.2]. The general case is solve... |

36 |
Moderately exponential bound for graph isomorphism
- BABAI
- 1981
(Show Context)
Citation Context ... elementary procedures for testing membership in Fd (see [Lul, §4]). For our purposes, it is essential only that d be fixed; the specific value of d would play a role in more precise timing arguments =-=[Ba2]-=-, [BL], [BKL]). The class Fd arose originally in the context of testing graph isomorphism ([Lul], [na2], [Mil], [Mi2], [eL], [FSS]). 3. Algorithmic preliminaries Unless indicated otherwise, subgroups ... |

32 |
L.: Canonical labelling of graphs in linear average time
- Babai, Kučera
- 1979
(Show Context)
Citation Context ...o graphs are isomorphic. In practice, ISO is not a hard problem (e.g., see [McK]). Indeed, on average over all graphs, and even over regular graphs, isomorphism is known to be testable in linear time =-=[BK]-=-, [Ku]. Furthermore, there is strong evidence that ISO is not NPcomplete, else the polynomial-time hierarchy would collapse to E~ = H E = AM ([GMW]). Nevertheless, ISO has stubbornly resisted attempts... |

28 |
An introduction to the group theory language, Cayley”, Computational group theory
- Cannon
- 1984
(Show Context)
Citation Context ...oblems apparently required access to quotients. 1. Introduction Since the order of a permutation group G on n letters can be exponential in n, it is customary, in both theory and practice (see, e.g., =-=[Ca]-=-, [FHL], [Si]), to specify G by a small set of generating permutations (less than n are needed and typically many fewer suffice). Despite the succinctness of such representations, a substantial polyno... |

27 | Monte carlo algorithms in graph isomorphism testing - Babai - 1979 |

27 |
Computational complexity and classification of finite simple groups
- BABAI, KANTOR, et al.
- 1983
(Show Context)
Citation Context ... [FHL], [Lull, [Mill, [Mi2], [BL]) used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group (=-=[BKL]-=-, [Lu2], [Ne], [KT], [Kal], [Ka2], [Ka3], [BLS1]), making available constructive versions of standard theoretical tools. 1 Research partially supported by NSF Grant DMS 87-01784 and NSA grant MDA 904-... |

27 |
Polynomial time algorithms for permutation groups
- Furst, Hopcroft, et al.
- 1980
(Show Context)
Citation Context ... apparently required access to quotients. 1. Introduction Since the order of a permutation group G on n letters can be exponential in n, it is customary, in both theory and practice (see, e.g., [Ca], =-=[FHL]-=-, [Si]), to specify G by a small set of generating permutations (less than n are needed and typically many fewer suffice). Despite the succinctness of such representations, a substantial polynomial-ti... |

25 |
On an algorithm for finding a base and strong generating set for a group given by generating permutations
- Leon
- 1980
(Show Context)
Citation Context ...t, in order to make use of this approach, it is necessary that elements of G be expressible as short words in )~. In fact, known algorithms for finding presentations of G admit this facility ([BLS1], =-=[Le]-=-), )~ appearing as a "strong generating set" of G. An algorithm for P2(ii) is a consequence of the straight-line (§3) construction of a presentation for P2(i). Duplicate the straight-line construction... |

25 | Computing the structure of finite algebras - Rónyai - 1990 |

24 | Sylow’s theorem in polynomial time
- Kantor
- 1985
(Show Context)
Citation Context ...) used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ([BKL], [Lu2], [Ne], [KT], [Kal], =-=[Ka2]-=-, [Ka3], [BLS1]), making available constructive versions of standard theoretical tools. 1 Research partially supported by NSF Grant DMS 87-01784 and NSA grant MDA 904-88-H-2040. 2 Research partially s... |

21 | Fast management of permutation groups I
- Babai, Luks, et al.
- 1997
(Show Context)
Citation Context ...ims's basic procedure ([Si]; see [FHL]) and the extension to G is trivial: IG/K[ = IGI/IKI. For K = 1, algorithms for P2(i) are standard (e.g., [LED; an asymptotically fast implementation is given in =-=[BLS2]-=-. Extensions to general K and procedures for P2(ii) follow easily from the nature of these presentations; see §12 for comments. Typical situations that we have in mind for H in P2(ii): H is input via ... |

21 | Computing the composition factors of a permutation group in polynomial time
- Luks
- 1987
(Show Context)
Citation Context ... [Lull, [Mill, [Mi2], [BL]) used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ([BKL], =-=[Lu2]-=-, [Ne], [KT], [Kal], [Ka2], [Ka3], [BLS1]), making available constructive versions of standard theoretical tools. 1 Research partially supported by NSF Grant DMS 87-01784 and NSA grant MDA 904-88-H-20... |

16 |
On the length of subgroup chains in the symmetric group
- Babai
- 1986
(Show Context)
Citation Context ... any strictly decreasing sequence of subgroups of Sym(n) has polynomial length (the bound log n! = O(n log n) is an immediate consequence of Lagrange's Theorem [Ha]; for the sharper bound 3n - 2, see =-=[Ba4]-=-). In this section, we recall a few fundamental problems for which polynomial-time algorithms are known. For these, there is no reasonable corresponding problem for quotient groups as the underlying s... |

16 | On the orders of primitive groups with restricted nonabelian composition factors - Babai, Cameron, et al. - 1982 |

15 |
Some algorithms for computing with finite permutation groups
- Neumann
- 1986
(Show Context)
Citation Context ... [Mill, [Mi2], [BL]) used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ([BKL], [Lu2], =-=[Ne]-=-, [KT], [Kal], [Ka2], [Ka3], [BLS1]), making available constructive versions of standard theoretical tools. 1 Research partially supported by NSF Grant DMS 87-01784 and NSA grant MDA 904-88-H-2040. 2 ... |

13 |
Isomorphism of k-contractible graphs. a generalization of bounded valence and bounded genus
- Miller
- 1983
(Show Context)
Citation Context ...at d be fixed; the specific value of d would play a role in more precise timing arguments [Ba2], [BL], [BKL]). The class Fd arose originally in the context of testing graph isomorphism ([Lul], [na2], =-=[Mil]-=-, [Mi2], [eL], [FSS]). 3. Algorithmic preliminaries Unless indicated otherwise, subgroups of Sym(n) = Sym(12) are input via generators. Output of groups is always via generators. All procedures identi... |

10 | Polynomial-time algorithms for finding elements of prime order and Sylow subgroups
- Kantor
- 1985
(Show Context)
Citation Context ...], [BL]) used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ([BKL], [Lu2], [Ne], [KT], =-=[Kal]-=-, [Ka2], [Ka3], [BLS1]), making available constructive versions of standard theoretical tools. 1 Research partially supported by NSF Grant DMS 87-01784 and NSA grant MDA 904-88-H-2040. 2 Research part... |

9 | Canonical Labeling of Regular Graphs in Linear Average Time - Kucera - 1987 |

7 | Specker E: Normal forms for trivalent graphs and graphs of bounded valence - Fürer, Schnyder - 1983 |

7 | Finding Sylow normalizers in polynomial time - KANTOR - 1990 |

7 | Polynomial-time versions of Sylow’s theorem
- Kantor, Taylor
- 1988
(Show Context)
Citation Context ..., [Mi2], [BL]) used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ([BKL], [Lu2], [Ne], =-=[KT]-=-, [Kal], [Ka2], [Ka3], [BLS1]), making available constructive versions of standard theoretical tools. 1 Research partially supported by NSF Grant DMS 87-01784 and NSA grant MDA 904-88-H-2040. 2 Resear... |

7 |
Isomorphism of graphs which are pairwise k-separable
- Miller
- 1983
(Show Context)
Citation Context ...chinery has developed for computing with permutation groups. A major stimulus for this activity was the application to the graph isomorphism problem (ISO), for early work ([Bal], [FHL], [Lull, [Mill, =-=[Mi2]-=-, [BL]) used groups to put significant instances of ISO into polynomial time. Ensuing studies resulted in algorithms for deciphering the basic building blocks of the group ([BKL], [Lu2], [Ne], [KT], [... |

5 |
NALJTY User’s Guide (Version 1.2
- MCKAY
- 1987
(Show Context)
Citation Context ...s. This is suggested by the fact that these are at least as hard as GRAPHISOMORPHISM (ISO), the problem of testing whether two graphs are isomorphic. In practice, ISO is not a hard problem (e.g., see =-=[McK]-=-). Indeed, on average over all graphs, and even over regular graphs, isomorphism is known to be testable in linear time [BK], [Ku]. Furthermore, there is strong evidence that ISO is not NPcomplete, el... |

4 | A Las Vegas-NC algorithm for isomorphism of graphs with bounded multiplicity of eigenvalues - Babai - 1986 |

4 |
Reduction of group constructions to point stabilizers
- COOPERMAN, FINKELSTEIN, et al.
- 1989
(Show Context)
Citation Context ...ing, from first principles, that the center of a permutation group is computable in polynomial-time involves only elementary properties of groups and no other knowledge of the group structure ([Lu2], =-=[CFL]-=-); it is, in fact, interpretable as the subgroup fixing a set of points (in an augmentation of the set) [Lu2] and so computable © 1990 ACM 089791-361-2/90/0005/0524 $1.50 524by the most basic algorit... |

2 | Some group-theoretic algorithms, Springer Lect - Sims - 1978 |

1 |
Algorithms for quotients of permutation groups
- Kantor, Luks
(Show Context)
Citation Context ...ribe efficient implementations. Of course, these are well-motivated, related issues, and each is the object of a growing literature. A more complete collection of algorithms and proofs will appear in =-=[KL]-=-. 2. Definitions and notation We recall some group-theoretical terminology. Throughout, let G be a finite group. We write H < G to indicate that H is a subgroup of G, and H <I G to indicate that H is ... |

1 |
A Las Vegas-NC algorithm for isomorphism of graphs with bounded multiplicity of eigenvalues
- BabaJ
- 1986
(Show Context)
Citation Context ...parts of P4 should be compared with the general problem INTERSECTION (see the Appendix), which is at least as hard as GRAPH-ISOMORPHISM (ISO) [Lua]. Problem P5 was previously open even for K = 1 (see =-=[Ba3]-=-). It is solved in §6. For K = 1, problem P6(i) is solved in [Lu2], while PT(i) is a consequence of P4(iii) and the computability of Csym(n)(B) (see, e.g., [CFL]). The general cases of P6(i), PT(i) ar... |

1 |
Canonical labeling of regular graphs in linear average time
- Kuera
- 1987
(Show Context)
Citation Context ...hs are isomorphic. In practice, ISO is not a hard problem (e.g., see [McK]). Indeed, on average over all graphs, and even over regular graphs, isomorphism is known to be testable in linear time [BK], =-=[Ku]-=-. Furthermore, there is strong evidence that ISO is not NPcomplete, else the polynomial-time hierarchy would collapse to E~ = H E = AM ([GMW]). Nevertheless, ISO has stubbornly resisted attempts to pl... |

1 |
Computing the structure of finite algebras
- R6nyai
- 1989
(Show Context)
Citation Context ...uence. On the other hand, el4(iii) is new even when K = 1 (cf. §6). An algorithm for P15(i) is discussed in §9. Computation of the "abelian part" of the socle requires an application of R6nyai's work =-=[R6]-=-. Plh(ii) is implicit in [BLS1]. We assume in P16 that ~ is specified by a, possibly parametrized, list of names of groups. We outline methods for these problems in §9. P16(ii) is actually implicit in... |