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This paper contributes to the generalization of Rademacher'sdifferentiability result for Lipschitz functions when the domain is infinitedimensional and has nonabelian group structure. We introduce the notion ofmetric scalable groups which are our infinite-dimensional analogues of Carnotgroups. The groups in which we will mostly be interested are the ones thatadmit a dense increasing sequence of (finite-dimensional) Carnot subgroups. Infact, in each of these spaces we show that every Lipschitz function has a pointof G\^{a}teaux differentiability. We provide examples and criteria for whensuch Carnot subgroups exist. The proof of the main theorem follows the work ofAronszajn and Pansu.

G\^ateaux differentiability on infinite-dimensional Carnot groups