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 ∩ U] as U ranges over all compact open subgroups of G. The properties of the scale function when passing to subgroups and quotient groups of G and under increasing unions of groups are also described.