We investigate the relationship between constant false alarm rate (CFAR) and invariant tests. We introduce the minimal invariant group (MIG). We show that for a family of distributions, the unknown parameters are eliminated from the distribution of the maximal invariant statistic under the MIG while the maximum information of the observed signal is preserved. We prove that any invariant test with respect to MIG is CFAR and conversely, for any CFAR test an invariant statistic exists with respect to an MIG under some mild conditions. Moreover for a given CFAR test, we propose a systematic method for deriving an enhanced test, i.e., a function of observations exists such that the likelihood ratio (LR) of the maximal invariant of its MIG gives an enhanced test. Furthermore, we introduce the uniformly most powerful-CFAR (UMP-CFAR) test as the optimal CFAR bound among all CFAR tests. We then prove that the UMP-CFAR test for the minimally invariant hypothesis testing problem is given by the LR of the maximal invariant under MIG. For some problems, this test (MP-CFAR) depends on the unknown parameters of the alternative hypothesis, however, provides an upper-performance bound for all suboptimal CFAR tests. We also propose three suboptimal novel CFAR tests among which one is asymptotically optimal.