Permutation Groups

Dixon, John D.; Mortimer, Brian

doi:10.1007/978-1-4612-0731-3

Cited by 1,123 publications

(1,158 citation statements)

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“…Remark 1. Lemma 1 is a well-known result in permutation group theory [11]. We give a proof in order to analyze the number of '3'-cycles which will be used in the decomposition process of our synthesis algorithm.…”

Section: Definition 4 (Even and Odd Permutations) A Permutation Is Ementioning

confidence: 96%

“…In this section, we introduce some basic concepts and results on permutation group theory from [11] and binary reversible logic from [12,13,14]. We introduce a permutation group and its relationship with reversible circuits.…”

Section: Preliminariesmentioning

confidence: 99%

“…We introduce a permutation group and its relationship with reversible circuits. [11]. A permutation group is simply a subgroup [11] of a symmetric group.…”

Section: Preliminariesmentioning

confidence: 99%

“…A mapping s : M → M can be written as a product of disjoint cycles (Definition 3) as an alternative notation for a mapping [11]. For example,…”

Section: Definition 2 (Permutation) Letmentioning

confidence: 99%

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Group Theory Based Synthesis of Binary Reversible Circuits

Yang

Song

Hung

et al. 2006

Lecture Notes in Computer Science

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Abstract. This paper presents an important result addressing a fundamental question in synthesizing binary reversible logic circuits for quantum computation. We show that any even-reversible-circuit of n (n > 3) qubits can be realized using NOT gate and Toffoli gate ('2'-ControlledNot gate), where the numbers of Toffoli and NOT gates required in the realization are bounded by (n+ n 3 )(3×2 2n−3 −2 n+2 ) and 4n(n+ n 3 )2 n , respectively. A provable constructive synthesis algorithm is derived. The time complexity of the algorithm is 10 3 n 2 · 2 n . Our algorithm is exponentially lower than breadth-first search based synthesis algorithms with respect to space and time complexities.

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Section: Definition 4 (Even and Odd Permutations) A Permutation Is Ementioning

confidence: 96%

Section: Preliminariesmentioning

confidence: 99%

“…We introduce a permutation group and its relationship with reversible circuits. [11]. A permutation group is simply a subgroup [11] of a symmetric group.…”

Section: Preliminariesmentioning

confidence: 99%

“…A mapping s : M → M can be written as a product of disjoint cycles (Definition 3) as an alternative notation for a mapping [11]. For example,…”

Section: Definition 2 (Permutation) Letmentioning

confidence: 99%

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Group Theory Based Synthesis of Binary Reversible Circuits

Yang

Song

Hung

et al. 2006

Lecture Notes in Computer Science

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“…We will consider the example given in [7, p. 468] (see also [4,Exercise 8.3.1]). Let p be a prime number.…”

Section: Properties Of Groups Satisfying the Cyclic-block Propertymentioning

confidence: 99%

Totally imprimitive permutation groups with the cyclic-block property

Asar¹

2011

Journal of Group Theory

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Abstract. Totally imprimitive p-groups satisfying the cyclic-block property are investigated. It is shown that in these groups any two blocks either are disjoint or one is contained in the other, and so the set of all blocks of the same size forms just one block system. Furthermore the non-FC-subgroups of these groups are transitive. For each prime p totally imprimitive p-subgroups of FSymðN Ã Þ satisfying the cyclic-block property are constructed, which are not minimal non-FC-groups.

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An infinite family of biprimitive semisymmetric graphs

Du¹,

Maru?i?²

1999

J. Graph Theory

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Permutation Groups

Cited by 1,123 publications

References 0 publications

Group Theory Based Synthesis of Binary Reversible Circuits

Group Theory Based Synthesis of Binary Reversible Circuits

Totally imprimitive permutation groups with the cyclic-block property

An infinite family of biprimitive semisymmetric graphs

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