Local analysis of the normalizer problem

✍ Scribed by Martin Hertweck

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 168 KB
• Volume: 163
• Category: Article
• ISSN: 0022-4049
• DOI: 10.1016/s0022-4049(00)00167-5

✦ Synopsis

For a ÿnite group G, and a commutative ring R, the automorphisms of G inducing an inner automorphism of the group ring RG form a group AutR(G). Let Aut int (G) = AutA(G), where A is the ring of all algebraic integers in C. It is shown how Cli ord theory can be used to analyze Aut int (G). It is proved that Aut int (G)=Inn(G) is an abelian group, and can indeed be any ÿnite abelian group. It is an outstanding question whether Aut Z (G) = Inn(G) if G has an abelian Sylow 2-subgroup. This is shown to be true in some special cases, but also a group G with abelian Sylow subgroups and Aut int (G) = Inn(G) is given.