Abstract. A totally disconnected, locally compact group G is said to be uniscalar if its scale function sG X G 3 N, as de®ned in [G. A. Willis, The structure of totally disconnected, locally compact groups, Math. Ann. 300 (1994), 341±363], is identically 1. 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. 1991 Mathematics Subject Classi®cation: 22E20; 20E08, 20F50. 1 Introduction Following Palmer , we say that a totally disconnected, locally compact group G is uniscalar if its scale function sG X G 3 N is identically 1, or, equivalently, if every element x e G normalizes some open, compact subgroup U of G (depending on x). This article is devoted to the study of uniscalar p-adic Lie groups. We are interested in the question whether the existence of the open, compact subgroups U normalized by individual group elements forces the existence of an open, compact
Forum Mathematicum – de Gruyter
Published: Apr 5, 2001
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