On a commutative non-associative algebra associated with a doubly transitive group
type: Item
instance of: scholarly article
title: On a commutative non-associative algebra associated with a doubly transitive group
zbMATH Open document ID: 0556.20002
DOI: 10.1016/0021-8693(84)90134-0
author: Koichiro Harada
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: Let G be a permutation group on \(\Omega =\{0,...,n\}\) and let V be the permutation module over a field K with basis \(\{v_ 0,...,v_ n\}\). Let U be the G-submodule spanned by \(v_ 0+...+v_ n\). Let A be the n- dimensional G-module V/U. Then A possesses at least one commutative algebraic structure such that the action induced by G on A yields that G is isomorphic to a subgroup of Aut(A). One such algebraic structure is given by the commutative algebra \(x_ ix_ i=(n-1)x_ i\), \(0\leq i\leq n\) and \(x_ ix_ j=-x_ i-x_ j\). If G is triply transitive on \(\Omega\), then up to scalar multiplication, this is the only possible nontrivial commutative structure. The present paper considers the case when G is doubly transitive on \(\Omega\) and all such possible algebra structures for A are found. The following question is also considered. Given G, can we find a G-invariant algebra structure for A such that Aut(A)\(\cong G?\) When \(G=L_ 3(2)\), an affirmative answer is given.
reviewed by: Ernest L. Stitzinger
Mathematics Subject Classification ID: 20B20
Mathematics Subject Classification ID: 17A05
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 3886106
zbMATH Keywords: permutation group
zbMATH Keywords: permutation module
zbMATH Keywords: doubly transitive
zbMATH Keywords: G-invariant algebra structure
zbMATH Keywords: \(L_ 3(2)\)
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90134-0
OpenAlex ID: W2047406201
cites work: Non-associative algebras associated with doubly transitive permutation groups: \(S_{n+1}\)
cites work: Q3963186
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description: scientific article; zbMATH DE number 3886106