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Like the lower central series of a nilpotent group, filters generalize theconnection between nilpotent groups and graded Lie rings. However, unlike thecase with the lower central series, the associated graded Lie ring may sharefew features with the original group: e.g. the associated Lie ring may betrivial or arbitrarily large. We determine properties of filters such that theLie ring and group are in bijection. We prove that, under such conditions,every isomorphism between groups is induced by an isomorphism between gradedLie rings.

Compatible filters with isomorphism testing