Computing automorphism groups and testing isomorphisms for modular group algebras.
The modular isomorphism problem of group algebras of finite \(p\)-groups over the prime field is still open. The aim of the paper is to provide a new algorithm for checking the existence of an isomorphism between the group algebras. There is already such an algorithm by \textit{M. Wursthorn}, [J. Symb. Comput. 15, No. 2, 211-227 (1993; Zbl 0782.20001)]. The new algorithm is effective and applicable for any nilponent finite dimensional algebra over a finite field. The elements of the algebra are identified with vectors of the coordinates, and multiplication is determined by the structure constant table. Two algebras are isomorphic if there is an invertible \(d\times d\) matrix compatible with multiplication. An isomorphism invariant table of canonical form is determined by induction according to the powers of the radical. The method uses the so-called covering algebra, analogous to the \(p\)-covering group of \(p\)-group algorithms due to \textit{M. F. Newman}, [Group Theory, Proc. Miniconf. Canberra 1975, Lect. Notes Math. 573, 73-84 (1977; Zbl 0519.20018)]. The input of the algorithm is the table of structure constants, the output is the canonical form, an isomorphism between these tables, and the generators and the order of the automorphism group. The algorithm was implemented in the computer algebra system GAP and was applied to check the isomorphism problem for groups of order \(2^8\) (more than 50000 in number) and \(3^6\) without yielding a counterexample. Previously the cases \(2^7\) and \(p^5\) had been checked.
scientific article; zbMATH DE number 5382249
scientific article