University of Sydney Algebra Seminar

Manoj K. Yadav (Harish-Chandra Research Institute)

Friday 13th September, 12:05-12:55pm, Carslaw 373

Class-preserving automorphisms and central quotients of finite \(p\)-groups

An automorphism of a given group is said to be (conjugacy) class-preserving if it maps each element of the group to a conjugate element. Notice that all inner automorphisms of any group are class-preserving. The first group having a non-inner class-preserving automorphism was constructed by W. Burnside in 1913, exactly 100 years ago. This group is a special \(p\)-group of order \(p^6\) for an odd prime \(p\), and it admits maximum possible number of class-preserving automorphisms. Starting from some historical remarks, I'll discuss about finite \(p\)-groups admitting maximum number of class-preserving automorphisms, and present somewhat surprising results when the nilpotency class of the group is larger than \(2\). A classification theorem will be presented in most of the cases. On the way, I'll introduce a generalization of Camina groups and report some interesting results. A Camina group is a group whose elements, lying in the complement of its commutator subgroup, have maximal conjugacy class size. A generalization of converse of a theorem of Schur by B. H. Neumann will be discussed, and a classification of finite \(p\)-groups having maximum possible central quotient will be presented as an application. For many other applications of class-preserving automorphisms and a geometric connection, please have a look at a recent article of Boris Kunyavskii availale at arXiv:1304.5053.