Generating groups using hypergraphs

Nick P Gill; Neil I Gillespie; Anthony Nixon; Jason P G Semeraro

doi:10.1093/qmath/haw001

Generating groups using hypergraphs

Nick P Gill, Neil I Gillespie, Anthony Nixon, Jason P G Semeraro

Research output: Contribution to journal › Article (Academic Journal) › peer-review

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Abstract

To a set B of 4-subsets of a set Ω of size n, we introduce an invariant called the ‘hole stabilizer’ which generalizes a construction of Conway, Elkies and Martin of the Mathieu group M12 based on Lloyd's ‘15-puzzle’. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs (Ω,B) with a trivial hole stabilizer, and determine all hole stabilizers associated to 2-(n,4,λ) designs with λ⩽2.

Original language	English
Pages (from-to)	29-52
Number of pages	24
Journal	Quarterly Journal of Mathematics
Volume	67
Issue number	1
DOIs	https://doi.org/10.1093/qmath/haw001
Publication status	Published - 1 Mar 2016

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title = "Generating groups using hypergraphs",

abstract = "To a set B of 4-subsets of a set Ω of size n, we introduce an invariant called the {\textquoteleft}hole stabilizer{\textquoteright} which generalizes a construction of Conway, Elkies and Martin of the Mathieu group M12 based on Lloyd's {\textquoteleft}15-puzzle{\textquoteright}. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs (Ω,B) with a trivial hole stabilizer, and determine all hole stabilizers associated to 2-(n,4,λ) designs with λ⩽2.",

author = "Gill, {Nick P} and Gillespie, {Neil I} and Anthony Nixon and Semeraro, {Jason P G}",

year = "2016",

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doi = "10.1093/qmath/haw001",

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T1 - Generating groups using hypergraphs

AU - Gill, Nick P

AU - Gillespie, Neil I

AU - Nixon, Anthony

AU - Semeraro, Jason P G

PY - 2016/3/1

Y1 - 2016/3/1

N2 - To a set B of 4-subsets of a set Ω of size n, we introduce an invariant called the ‘hole stabilizer’ which generalizes a construction of Conway, Elkies and Martin of the Mathieu group M12 based on Lloyd's ‘15-puzzle’. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs (Ω,B) with a trivial hole stabilizer, and determine all hole stabilizers associated to 2-(n,4,λ) designs with λ⩽2.

AB - To a set B of 4-subsets of a set Ω of size n, we introduce an invariant called the ‘hole stabilizer’ which generalizes a construction of Conway, Elkies and Martin of the Mathieu group M12 based on Lloyd's ‘15-puzzle’. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs (Ω,B) with a trivial hole stabilizer, and determine all hole stabilizers associated to 2-(n,4,λ) designs with λ⩽2.

U2 - 10.1093/qmath/haw001

DO - 10.1093/qmath/haw001

M3 - Article (Academic Journal)

SN - 0033-5606

VL - 67

SP - 29

EP - 52

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

IS - 1

ER -

Generating groups using hypergraphs

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