The GroupLASSO method for finding important explanatory factors suffers from the potential non-uniqueness of solutions and also from high computational costs. We formulate conditions for the uniqueness of GroupLASSO solutions which lead to an easily implementable test procedure. In addition to merely detecting ambiguities in solutions, this testing procedure identifies all potentially active groups. These results are used to derive an efficient algorithm that can deal with input dimensions in the millions and can approximate the solution path efficiently. The derived methods are applied to large-scale learning problems where they exhibit excellent performance. We show that the proposed testing procedure helps to avoid misinterpretations of GroupLASSO solutions.