An elementary proof that classical braids embed in virtual braids

The purpose of this paper is to prove that the natural mapping of classical braids to virtual braids is an embedding. The proof does not use any complete invariants of classical braids; it is based on a projection from (colored) virtual braids onto classical braids (which is similar to the projection in [6]); this projection is the identity mapping on the set of classical braids. It is well defined do not only for the group of (colored) virtual braids but also for the quotient group of the group of (colored) virtual braids by the so-called virtualization motion. The idea of this projection is closely related to the notion of parity and the groups G (n) (k) introduced by the author in [3].