On schur rings over cyclic groups

Leung, Ka Hin; Man, Shing Hing

doi:10.1007/bf02773471

Cited by 58 publications

(93 citation statements)

References 6 publications

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“…Thus S and S * are isomorphic S-rings. Now there is a classification of S-rings over cyclic groups found in [LManII;Theorem 3.7] which describes such an S-ring S * as being of one of four Types:…”

Section: Background and Further Resultsmentioning

confidence: 99%

Fusions of character tables III: Fusions of cyclic groups and a generalisation of a condition of Camina

2010

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In this paper we characterise finite groups whose character tables fuse from the character table of a cyclic group. We also show a connection with a generalisation of the Camina pair condition introduced by Camina in [Ca]. IntroductionIn this paper we characterise finite groups whose character tables fuse from the character table of a cyclic group. We also show a connection with a generalisation of a Camina pair (called a Camina triple) introduced in [Ca]; see [DY, CMS, K, MS, Ma] for some results on Camina pairs. We will give two other characterisations of Camina triples, each of which is character-theoretic and which mirror known characterisations of Camina pairs.

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Section: Background and Further Resultsmentioning

confidence: 99%

Fusions of character tables III: Fusions of cyclic groups and a generalisation of a condition of Camina

2010

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“…By statement (1) with S = ∅ there exists a character χ ∈R × such that χ (S) = χ (S ) = 0. However, due to (11) we have χ = χ (r) for some r ∈ R × . Thus, χ (r S) = χ (S) = 0.…”

Section: Theorem 41 Let R Be a Galois Ring Of Characteristic P N Anmentioning

confidence: 97%

Schur rings over a Galois ring of odd characteristic

Evdokimov

Ponomarenko

2010

Journal of Combinatorial Theory, Series A

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“…Since g is conjugate in Aut(W ) to g , we are done. Now we prove the existence of h * ∈ Aut(W ) satisfying (13). For this, we observe that the permutation g l with l = |V/E 1 | induces the identical permutation of X/E 1 and a full cycle of X/E 2 .…”

Section: Theorem 42 In the Above Notation For Any Family F The Gromentioning

confidence: 99%

“…This algorithm allows us to efficiently find a cycle base of an arbitrary solvable permutation group. The consistency of the Main Algorithm follows from Theorem 5.1 based on deep results on the structure of Schur rings over a cyclic group [11,13].…”

Section: Theorem 12 Let G N (Respectively C N ) Be the Class Of Almentioning

confidence: 99%

Circulant graphs: recognizing and isomorphism testing in polynomial time

Evdokimov

Ponomarenko

2004

St. Petersburg Math. J.

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Abstract. An algorithm is constructed for recognizing the circulant graphs and finding a canonical labeling for them in polynomial time. This algorithm also yields a cycle base of an arbitrary solvable permutation group. The consistency of the algorithm is based on a new result on the structure of Schur rings over a finite cyclic group. §1. Introduction A finite graph1 is said to be circulant if its automorphism group contains a full cycle, i.e., a permutation the cycle decomposition of which consists of a unique cycle of full length. This means that the graph admits a regular cyclic automorphism group, and, consequently, is isomorphic to a Cayley graph over a cyclic group. In particular, any circulant graph can be specified in a compact form by a full cycle automorphism and a neighborhood of some vertex. One of the main computational problems concerning circulant graphs is that of finding an efficient algorithm to recognize them. (This problem is a special case of the following NP-complete problem: test whether or not a given graph has an automorphism without fixed points [15].) The first attempt to solve this problem was undertaken in [24], where a polynomial-time algorithm for recognizing circulant tournaments was described. In the subsequent papers [21,22,5] several results on recognizing some special classes of circulant graphs were presented, but the general problem remained open up to now. In the present paper we solve this problem completely. Another problem about circulant graphs is to find an efficient isomorphism test for them. In fact, this problem is polynomial-time reducible to the recognition problem, because two circulant graphs with the same number of vertices are isomorphic if and only if their disjoint union is a circulant graph. In this paper we present a solution to a more difficult problem of finding a canonical labeling for circulant graphs.3 It should be mentioned that the isomorphism problem for Cayley graphs over a cyclic group (which is a special case of the isomorphism problem for circulant graphs) has been extensively studied through the last forty years (see [20]

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On schur rings over cyclic groups

Cited by 58 publications

References 6 publications

Fusions of character tables III: Fusions of cyclic groups and a generalisation of a condition of Camina

Fusions of character tables III: Fusions of cyclic groups and a generalisation of a condition of Camina

Schur rings over a Galois ring of odd characteristic

Circulant graphs: recognizing and isomorphism testing in polynomial time

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