The Series of Norms in a Soluble p-Group

The norm of a group G is the subgroup of elements of G which normalise every subgroup of G . We shall denote it κ( G ). An ascending series of subgroups κ i ( G ) in G may be defined recursively by: κ 0 ( G ) = 1 and, for i [ges ]0, κ i +1 ( G )/κ i ( G ) = κ( G /κ i ( G )). For each i , the section κ i +1 ( G )/κ i ( G ) clearly contains the centre of the group G /κ i ( G ). A result of Schenkman [ 8 ] gives a very close connection between this norm series and the upper central series: ζ i ( G )⊆κ i ( G ) ⊆ζ 2 i ( G ).