The Series of Norms in a Soluble p -Group
R. A. Bryce, John Cossey · Bulletin of the London Mathematical Society · 1997
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 κi(G) in G may be defined recursively by: κ0(G) = 1 and, for i ⩾ 0, κi+1(G)/κi(G) = κ(G/κi(G)). For each i, the section κi+1(G)/κi(G) clearly contains the centre of the group G/κi(G). A result of Schenkman [8] gives a very close connection between this norm series and the upper central series: ζi(G) ⊆ κi(G) ⊆ ζ2i(G). 1991 Mathematics Subject Classification 20E15.