Symmetric groups as automorphisms of some commutative algebras
Let \(A\) be a commutative, nonassociative algebra over a field \(k\) such that \(A\) has a basis \(x_ 1,...,x_ n\), \(n\ge 2\), such that \(x_i x_j=-x_i-x_j\) for \(1\leq i<j\leq n\) and \(x_i x_i=(n-1)x_i\) for \(1\leq i\leq n\). Let \(p=\text{char}\, k\) and \(\Aut(A)\) be the automorphism group of \(A\). \(\Aut(A)\) has been determined by the author [J. Fac. Sci., Univ. Toyko, Sect. I A 28, 843--849 (1981; Zbl 0498.17001)] and \textit{H. P. Allen} [J. Algebra 91, 265--279 (1984; Zbl 0556.20004)]. The present paper gives a simple proof of their result which follows: \(\Aut(A)=\Sigma_{n+1}\ \text{if}\ p=0\ \text{or}\ p \nmid n+1\ \text{and} \Aut(A)=k^{n-1}\cdot \mathrm{GL}(n-1,k)\quad \text{if}\ p\mid n+1\), where \(\Sigma_{n+1}\) is the symmetric group of degree \(n+1\) and \(k^{n-1} \mathrm{GL}(n-1,k)\) is a natural extension of an \(n-1\) dimensional vector space \(k^{n-1}\) by \/ \mathrm{GL}(n-1,k)\).
scientific article; zbMATH DE number 3955157