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Intense relativistic electron beams have been injected into the plasma of a Z pinch at various times during the formation of the plasma column. If the diameter of the plasma column exceeded a certain value, these beams were found to propagate with a high transport efficiency when injected either into the collapsing current sheet of the Z pinch or into the expanding plasma column after its diameter had reached a minimum. However, when the diameter of the plasma column is smaller than this critical value, the relativistic electron beam does not propagate. Instead, it continues to diverge until it strikes the glass wall of the Z pinch tube.