The unit group of \(FS_4\).
Let \(FS_m\) denote a group algebra of the symmetric group \(S_m\) of degree \(m\) over a finite field \(F\). In [Acta Math. Acad. Paedagog. Nyházi. (N.S.) 23, No. 2, 129-142 (2007; Zbl 1135.16034)] the authors characterized the unit group \(U(FS_3)\) of \(FS_3\). This work is continued here, they consider the group \(S_4\) and prove that if \(F\) is the field of \(p^k\) elements, where \(p>3\), then the unit group \(U(FS_4)\) is isomorphic to \[ \text{GL}(3,F)\times\text{GL}(3,F)\times\text{GL}(2,F)\times U(F)\times U(F), \] where \(\text{GL}(n,F)\) denotes the general linear group of degree \(n\) over \(F\). Furthermore, if \(p\leq 3\), then \(U(FS_4)\) is an extension of the group \(G\) by the group \(1+J(FS_4)\), where \(G\) is either \(\text{GL}(2,F)\times U(F)\) or \(\text{GL}(3,F)\times\text{GL}(3,F)\times U(F)\times U(F)\) depending on whether \(p\) is equal to \(2\) or \(3\), and \(J(FS_4)\) is the Jacobson radical of \(FS_4\). Also the structure of \(1+J(FS_4)\) is determined in both cases, and the characterization is hereby complete.
scientific article; zbMATH DE number 5355075
scientific article