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In this paper, we address the problem of adaptive detection in homogeneous Gaussian interference, with unknown covariance matrix, after reduction by invariance. Starting from a maximal invariant statistic, which contains all the information for the synthesis of an invariant detector, we devise the Rao test, generalized likelihood ratio test (GLRT), and Durbin test. Moreover, we compare their decision statistics with those of the receivers designed according to the same criteria from the raw data (i.e., before reduction by invariance). We prove that the GLRT in the original data space is statistically equivalent to the GLRT designed after reduction by invariance (under a very mild assumption) and coincide with the conditional uniformly most powerful invariant (C-UMPI) test, obtained conditioning on an ancillary part of the maximal invariant statistic. As to the Rao and Durbin criteria, when they are applied after reduction by invariance, lead to detectors different from the counterparts devised in the raw data domain. At the analysis stage, the performance of the receivers devised in the invariant domain is analyzed in comparison with that of the counterparts synthesized from the original observations.