We report the observation of Shubnikov-de Haas oscillations in the underdoped cuprate superconductor YBa$_2$Cu$_4$O$_8$ (Y124). For field aligned along the c-axis, the frequency of the oscillations is $660pm 30$ T, which corresponds to $sim 2.4$ % of the total area of the first Brillouin zone. The effective mass of the quasiparticles on this orbit is measured to be $2.7pm0.3$ times the free electron mass. Both the frequency and mass are comparable to those recently observed for ortho-II YBa$_2$Cu$_3$O$_{6.5}$ (Y123-II). We show that although small Fermi surface pockets may be expected from band structure calculations in Y123-II, no such pockets are predicted for Y124. Our results therefore imply that these small pockets are a generic feature of the copper oxide plane in underdoped cuprates.
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