Abelian Steiner Triple Systems

Tannenbaum, Peter

doi:10.4153/cjm-1976-124-6

Cited by 9 publications

(5 citation statements)

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“…For v = 25, the CycleStructurePerm command allows us to see that if G is not an abelian group of order 25 then G possesses at least one permutation g satisfying one of the following properties: It is known that for v = 25 there is a sharply transitive STS(25) with respect to G, when G is an abelian group of order 25, [19]. By Proposition 14 an STS(25) which is invariant under a sharply transitive abelian group G of order 25 yields a bowtie decomposition which is sharply transitive with respect to G. Hence, for v = 25 the only examples of transitive bowtie decompositions are the sharply transitive decompositions under one of the abelian groups of order 25 (cyclic or elementary abelian).…”

Section: The Electronic Journal Of Combinatorics 17 (2010) #R101mentioning

confidence: 99%

“…For the other admissible values of v, the existence of a sharply transitive decomposition B v turns out to be equivalent to the existence of a sharply transitive STS(v) (see Proposition 14). In this way, a large class of examples for sharply transitive bowtie decompositions of K v , with v ≡ 1 (mod 12), can be obtained from the abelian STS(v)'s constructed in [19].…”

Section: Introductionmentioning

confidence: 99%

“…As already mentioned, for every v ≡ 1 (mod 12), a large class of sharply transitive bowtie decompositions of K v can be obtained from the abelian STS(v)'s which have been constructed in [19]. In particular, for every admissible value of v there exists a sharply transitive bowtie decomposition of K v with respect to the cyclic group of order v.…”

mentioning

confidence: 99%

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Symmetric Bowtie Decompositions of the Complete Graph

Bonvicini¹,

Ruini²

2010

Electron. J. Combin.

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Given a bowtie decomposition of the complete graph $K_v$ admitting an automorphism group $G$ acting transitively on the vertices of the graph, we give necessary conditions involving the rank of the group and the cycle types of the permutations in $G$. These conditions yield non–existence results for instance when $G$ is the dihedral group of order $2v$, with $v\equiv 1, 9\pmod{12}$, or a group acting transitively on the vertices of $K_9$ and $K_{21}$. Furthermore, we have non–existence for $K_{13}$ when the group $G$ is different from the cyclic group of order $13$ or for $K_{25}$ when the group $G$ is not an abelian group of order $25$. Bowtie decompositions admitting an automorphism group whose action on vertices is sharply transitive, primitive or $1$–rotational, respectively, are also studied. It is shown that if the action of $G$ on the vertices of $K_v$ is sharply transitive, then the existence of a $G$–invariant bowtie decomposition is excluded when $v\equiv 9\pmod{12}$ and is equivalent to the existence of a $G$–invariant Steiner triple system of order $v$. We are always able to exclude existence if the action of $G$ on the vertices of $K_v$ is assumed to be $1$–rotational. If, instead, $G$ is assumed to act primitively then existence can be excluded when $v$ is a prime power satisfying some additional arithmetic constraint.

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Section: The Electronic Journal Of Combinatorics 17 (2010) #R101mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetric Bowtie Decompositions of the Complete Graph

Bonvicini¹,

Ruini²

2010

Electron. J. Combin.

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“…253-255. We are grateful to D. G. Rogers (private communication) for pointing out this result as informing us of the following 3-regular complete mapping of Z 2h \ φ = (1,8,5) (2,10,11) (3,6,24) (4,14,16) (7,19,17) (9,15,20) (12, 23, 18) (13, 22, 21). The above partition was obtained by D. G. Rogers and F. W. Roush by means of a computer search.…”

Section: Note Added In Proofmentioning

confidence: 99%

“…Π For the case k = 6, a 6-regular complete mapping of Z Λ for WΞI (mod 6) can be constructed from a C/P-neofield N v of order v = 2 (mod 6) [1] or from an £ΓP /-matrix of order m Ξ 0 (mod 6) [8]. This result can be extended to show the existence of a 6-regular complete mapping of any abelian group of order = 1 (mod 6) [14]. 4* A related number theoretic problem* Let φ: G -> G be a complete mapping of G, normalized (as in the introduction) so that φ fixes the identity element of G. Then as already noted, φ can be regarded as a permutation ( 1 2 n ) of the elements of G with the property that α έ ~h~i 1 c i (i = 1, , w -1) also constitute all the nonidentity elements of G. We now decompose this permutation into a product of disjoint cycles, and suppose that (b 1 b 2 b r ) is a typical one of these cycles.…”

mentioning

confidence: 99%