On a commutative non-associative algebra associated with a doubly transitive group
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.
scientific article; zbMATH DE number 3886106