Normalizes and permutational isomorphisms in simply-exponential time

Wiebking, Daniel

doi:10.1137/1.9781611975994.14

Cited by 11 publications

(10 citation statements)

References 13 publications

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“…For a long time, it was not even known whether the normaliser problem could be solved in time that was simply exponential in n, since a naive algorithm involves looking through up to all 2 n log n elements of S n for normalising elements. However, Wiebking in (96) has recently shown that the normaliser problem can be solved in time 2 O(n) .…”

Section: Graph Isomorphism and Related Problemsmentioning

confidence: 99%

Maximal subgroups of finite simple groups: classifications and applications

Roney-Dougal¹

2021

Surveys in Combinatorics 2021

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This paper surveys what is currently known about the maximal subgroups of the finite simple groups. After briefly introducing the groups themselves, if their maximal subgroups are completely determined then we present this classification. For the remaining finite simple groups our current knowledge is only partial: we describe the state of play, as well as giving some results that apply more generally. We also direct the reader towards computational resources for the construction of maximal subgroups.After this, we present three sample applications, selected because they combine group theoretical and combinatorial arguments, and because they use either or both of the detailed classifications and the looser statements that can be made about all maximal subgroups. In particular, we discuss results relating to generation, and the generating graph; results concerning bases; and some applications to computational complexity, in particular to graph colouring and other problems with no known polynomial-time solution.

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Section: Graph Isomorphism and Related Problemsmentioning

confidence: 99%

Maximal subgroups of finite simple groups: classifications and applications

Roney-Dougal¹

2021

Surveys in Combinatorics 2021

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“…In this paper, we investigate the theoretical complexity of the normaliser problem. It is shown in [13] that the problem can be solved in simply exponential time 2 O(n) , but there is no known subexponential solution to the general problem, and in fact, the fastest practical algorithms all use a backtrack search whose worst-case complexity is greater than exponential. A permutation group problem P is said to be quasipolynomial if there exists a constant c such that P can be solved in time 2 O(log c n) , where n is the degree of the underlying group or groups.…”

Section: Introductionmentioning

confidence: 99%

Primitive normalisers in quasipolynomial time

Chang

Roney-Dougal

2021

Arch. Math.

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The normaliser problem has as input two subgroups H and K of the symmetric group $$\mathrm {S}_n$$ S n , and asks for a generating set for $$N_K(H)$$ N K ( H ) : it is not known to have a subexponential time solution. It is proved in Roney-Dougal and Siccha (Bull Lond Math Soc 52(2):358–366, 2020) that if H is primitive, then the normaliser problem can be solved in quasipolynomial time. We show that for all subgroups H and K of $$\mathrm {S}_n$$ S n , in quasipolynomial time, we can decide whether $$N_{\mathrm {S}_n}(H)$$ N S n ( H ) is primitive, and if so, compute $$N_K(H)$$ N K ( H ) . Hence we reduce the question of whether one can solve the normaliser problem in quasipolynomial time to the case where the normaliser in $$\mathrm {S}_n$$ S n is known not to be primitive.

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“…Given generators for subgroups G and H of the symmetric group S n , the normaliser problem asks one to compute a generating set for the normaliser N G (H). The fastest known bound for this problem, in general, is Wiebking's simply exponential bound 2 O(n) [20]. Better bounds are known for various cases: for example, quasipolynomial 2 O(log 3 n) if N Sn (H) is primitive [16,5], and polynomial if G has restricted composition factors by work of Luks and Miyazaki [13].…”

Section: Introductionmentioning

confidence: 99%

“…As Wiebking's simply exponential algorithm [20] is not implemented, the fastest implemented algorithm to compute N Sn (H) has a runtime complexity of 2 O(n log n) . Using methods based on the work of Feulner [6], we shall bound the complexity of Norm-Sym for H ∈ InP(C p ) by 2…”

Section: Introductionmentioning

confidence: 99%

Computing normalisers of intransitive groups

Chang¹,

Jefferson²,

Roney-Dougal³

2021

Preprint

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The normaliser problem takes as input subgroups G and H of the symmetric group S n , and asks one to compute N G (H). The fastest known algorithm for this problem is simply exponential, whilst more efficient algorithms are known for restricted classes of groups. In this paper, we will focus on groups with many orbits. We give a new algorithm for the normaliser problem for these groups that performs many orders of magnitude faster than previous implementations in GAP. We also prove that the normaliser problem for the special case G = S n is at least as hard as computing the group of monomial automorphisms of a linear code over any field of fixed prime order.

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Normalizes and permutational isomorphisms in simply-exponential time

Abstract: We show that normalizers and permutational isomorphisms of permutation groups given by generating sets can be computed in time simply exponential in the degree of the groups. The result is obtained by exploiting canonical forms for permutation groups (up to permutational isomorphism).

Cited by 11 publications

References 13 publications

Maximal subgroups of finite simple groups: classifications and applications

Maximal subgroups of finite simple groups: classifications and applications

Primitive normalisers in quasipolynomial time

Computing normalisers of intransitive groups

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