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A group theory justification of one dimensional fractional supersymmetry is proposed using an analogue of a coset space, just like the one introduced in 1D supersymmetry. This theory is then gauged to obtain a local fractional supersymmetry i.e. a fractional supergravity which is then quantized à la Dirac to obtain an equation of motion for a particle which is in a representation of the braid group and should describe alternative statistics. A formulation invariant under generaldoi:10.1142/s0217732396000916 fatcat:bvatxholwfeyrorn7emhfrmo3i