Polynomial-time normalizers for permutation groups with restricted composition factors

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For an integer constant d \textgreater 0, let Gamma(d) denote the class of finite groups all of whose nonabelian composition factors lie in S-d; in particular, Gamma(d) includes all solvable groups. Motivated by applications to graph-isomorphism testing, there has been extensive study of the complexity of computation for permutation groups in this class. In particular, the problems of finding set stabilizers, intersections and centralizers have all been shown to be polynomial-time computable. A notable open issue for the class Gamma(d) has been the question of whether normalizers can be found in polynomial time. We resolve this question in the affirmative. We prove that, given permutation groups G, H \textless= Sym(Omega) such that G is an element of Gamma(d), the normalizer of H in G can be found in polynomial time. Among other new procedures, our method includes a key subroutine to solve the problem of finding stabilizers of subspaces in linear representations of permutation groups in Gamma(d).

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Permutation groups and orbits on the power set

Journal of Algebra and Its Applications ◽

10.1142/s0219498821500055 ◽

2019 ◽

Vol 19 (12) ◽

pp. 2150005

Author(s):

Yong Yang

Keyword(s):

Permutation Group ◽

Permutation Groups ◽

Power Set ◽

Composition Factors

Let [Formula: see text] be a permutation group of degree [Formula: see text] and let [Formula: see text] denote the number of set-orbits of [Formula: see text]. We determine [Formula: see text] over all groups [Formula: see text] that satisfy certain restrictions on composition factors.

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Permutation groups and polynomial-time computation

DIMACS Series in Discrete Mathematics and Theoretical Computer Science - Groups and Computation ◽

10.1090/dimacs/011/11 ◽

1993 ◽

pp. 139-175 ◽

Cited By ~ 42

Author(s):

Eugene Luks

Keyword(s):

Polynomial Time ◽

Permutation Groups

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ON THE COMPLEXITY OF THE HIDDEN SUBGROUP PROBLEM

International Journal of Foundations of Computer Science ◽

10.1142/s0129054113500305 ◽

2013 ◽

Vol 24 (08) ◽

pp. 1221-1234 ◽

Cited By ~ 2

Author(s):

STEPHEN FENNER ◽

YONG ZHANG

Keyword(s):

Quantum Computing ◽

Computational Complexity ◽

Polynomial Time ◽

Permutation Groups ◽

Dihedral Groups ◽

Hidden Subgroup Problem ◽

Subgroup Problem ◽

The Relationship ◽

Decision Version

We study the computational complexity of the HIDDEN SUBGROUP problem, a well-studied problem in quantum computing. First we show that several proposed generalizations or variants of this problem, including HIDDEN COSET, HIDDEN SHIFT, and ORBIT COSET, are all equivalent or reducible to HIDDEN SUBGROUP. Then we study the relationship between the decision version and search version of HIDDEN SUBGROUP over various group classes. We show that the two versions are polynomial-time equivalent over permutation groups, and over dihedral groups given the order of the group is smooth. Finally, we give nonadaptive program checkers for HIDDEN SUBGROUP and its decision version.

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Seminormal and subnormal subgroup lattices for transitive permutation groups

Journal of the Australian Mathematical Society ◽

10.1017/s144678870001137x ◽

2006 ◽

Vol 80 (1) ◽

pp. 45-64

Author(s):

Cheryl E. Praeger

Keyword(s):

Subnormal Subgroup ◽

Subgroup Lattice ◽

Permutation Groups ◽

Transitive Permutation Group ◽

Transitive Groups ◽

Lattices Of Subgroups ◽

Subgroup Lattices ◽

A Chain ◽

Seminormal Subgroup ◽

Composition Factors

AbstractVarious lattices of subgroups of a finite transitive permutation group G can be used to define a set of ‘basic’ permutation groups associated with G that are analogues of composition factors for abstract finite groups. In particular G can be embedded in an iterated wreath product of a chain of its associated basic permutation groups. The basic permutation groups corresponding to the lattice L of all subgroups of G containing a given point stabiliser are a set of primitive permutation groups. We introduce two new subgroup lattices contained in L, called the seminormal subgroup lattice and the subnormal subgroup lattice. For these lattices the basic permutation groups are quasiprimitive and innately transitive groups, respectively.

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