Further properties of the scale function on a totally disconnected group

A new characterisation of the scale function on the locally compact group G is given. It is shown that for x in G the scale of x, s(x), is the minimum value attained by the index [xUx(-1): xUx(-1) boolean AND U] as U ranges o ver all compact open subgroups of G. The properties of the scale function w hen passing to subgroups and quotient groups of G and under increasing unio ns of groups are also described. (C) 2001 Academic Press.