The transitive groups of degree twelve
type: Item
instance of: scholarly article
title: The transitive groups of degree twelve
zbMATH Open document ID: 0683.20002
DOI: 10.1016/S0747-7171(87)80068-8
published in: Journal of Symbolic Computation
publication date: 1987-01-01T00:00:00Z
review text: The author describes an elementary method for constructing all the pairwise nonconjugate transitive groups of degree 12. The method has been implemented by the author using the group theory package CAYLEY, yielding a list of 301 transitive groups of degree 12. Details are also given of the 17 groups in this list which are minimal transitive. The author compares his results with those of \textit{G. A. Miller} [Q. J. Math. 28, 193-231 (1896; JFM 27.0097.03)], who also gave a list of transitive groups of degree 12. Miller's list differs from the author's in surprisingly few places. The basic method is as follows. Let \(G\) be a transitive group of degree 12. If \(G\) is primitive then \(G\) is one of the groups \(PSL_ 2(11)\), \(PGL_ 2(11)\), \(M_{11}\), \(M_{12}\), \(A_{12}\) or \(S_{12}\) [see \textit{C. C. Sims}, in Comput. Probl. abstract Algebra, Oxford 1967, 169-183 (1970; Zbl 0215.10002)], so one may suppose that \(G\) is imprimitive, with a block system consisting of \(c\) blocks of size \(d\), where \(cd=12\). The various values of \(c\) are dealt with case by case, the method being to identify the possibilities for the kernel \(K\) of the action of \(G\) on the block system. Then \(G/K\leq N(K)/K\), and the group \(N(K)/K\) is usually small enough for the subgroup lattice programme in CAYLEY to be applied to give a list of possibilities for \(G/K\), and hence for \(G\).
Mathematics Subject Classification ID: 20B05
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 20-04
Mathematics Subject Classification ID: 20B15
zbMATH DE Number: 4118567
zbMATH Keywords: transitive groups of degree 12
zbMATH Keywords: CAYLEY
zbMATH Keywords: minimal transitive
zbMATH Keywords: primitive
zbMATH Keywords: block system
zbMATH Keywords: action
author: Gordon Royle
describes a project that uses: Cayley
MaRDI profile type: Publication
cites work: Computing normalizers in permutation groups
cites work: The transitive groups of degree up to eleven+
cites work: Q3333220
cites work: Q5617749
cites work: Q5512231
description: scientific article; zbMATH DE number 4118567