  We study problems of maximal symmetry in Banach spaces. This is done by
providing an analysis of the structure of small subgroups of the general linear
group GL(X), where X is a separable reflexive Banach space. In particular, we
provide the first known example of a Banach space X without any equivalent
maximal norm, or equivalently such that GL(X) contains no maximal bounded
subgroup. Moreover, this space X may be chosen to be super-reflexive.
