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We introduce generalized presentations of groups. Roughly speaking, a generalized presentation of a group \(G\) consists of a generalized free group \(\mathcal{F}\) (which is a certain subgroup of a big free group \({\rm BF(\Lambda)}\)) and of a subset \(R\) of \(\mathcal{F}\) such that \(G\) is isomorphic to \(\mathcal{F}/\overline{\langle\langle R\rangle\rangle}\), where \(\overline{\langle\langle R\rangle\rangle}\) is
the closure of \(\langle\langle R\rangle\rangle\) with respect to an appropriate topology on \(\mathcal{F}\).
We give a generalized presentation of \({\rm Aut}(F_{\omega})\), the automorphism group of the free group
of infinite countable rank. This generalized presentation is countable,
although the group itself is uncountable.
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