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We prove a new interesting property of space-time block codes (STBCs) that are based on generalized orthogonal designs. For flat block-fading channels, it is shown that the internal structure of the vector space of the input constellation remains invariant to the combined effect of the STBC and the channel, except for a scaling factor. The established constellation space invariance property is entirely due to the specific structure of the STBCs based on the generalized orthogonal designs. Using this property, we obtain simple exact expressions for the error probability of the maximum likelihood (ML) decoder in the general case when the channel, STBC, and input signal constellations are arbitrary. Such expressions are obtained in both the cases when the channel realization is deterministic (fixed) and random. In the latter case, simple expressions are derived for the average error probability.