Computing automorphism groups of finite groups
This paper gives an outline (a longer version is promised) of a new algorithm for computing the automorphism group of a finite group. Similar to the author's algorithm for subgroups [\textit{J. J. Cannon, B. C. Cox, D. F. Holt}, J. Symb. Comput. 31, No. 1-2, 149-161 (2001; Zbl 0984.20002)] it uses tabulated information to obtain the automorphism group for the factor of the largest normal solvable subgroup, and then lifts the result via steps over elementary Abelian chief factors to the whole group.NEWLINENEWLINENEWLINEThis lifting step generalizes the method for \(p\)-groups of \textit{E. A. O'Brien} [Computational algebra and number theory, Math. Appl., Dordr. 325, 83-90 (1995; Zbl 0836.20002)] and the method for solvable groups of \textit{M. J. Smith} [ Doctoral Thesis, Australian National University (1994)].NEWLINENEWLINENEWLINEThe main calculations concern the automorphism group of \(KG\)-modules, a potential bottleneck is to determine which group automorphisms induce particular module isomorphisms.NEWLINENEWLINENEWLINEThe paper closes with a list of performance figures that show that the method proposed is certainly practical for groups of size \(10^6\) (and likely larger groups as well). In fact with a few obvious exceptions (mainly \(p\)-groups) it succeeded on a list of all transitive groups of degree up to 30 determined by the reviewer [Konstruktion transitiver Permutationsgruppen, Aachener Beiträge zur Mathematik 18 (1996; Zbl 0955.20002)].NEWLINENEWLINEFor the entire collection see [Zbl 0959.00030].
scientific article; zbMATH DE number 1683593