Uniscalar p-adic Lie groups

A totally disconnected, locally compact group G is said to be uniscalar if its scale function SG : G --> N, as defined in [G. A. Willis, The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), 341-363], is identically I. It is known that G is uniscalar if and only if every element of G normalizes some open, compact subgroup of G. We show that every identity neighbourhood of a compactly generated, uniscalar p-adic Lie group contains an open, compact, normal subgroup. In contrast, uniscalar p-adic Lie groups which are not compactly generated need not possess open, compact, normal subgroups.