  We propose a group theoretic condition which may be applied to extensions of
the Standard Model in order to locate regions of parameter space in which the
electroweak phase transition is strongly first order, such that electroweak
baryogenesis may be a viable mechanism for generating the baryon asymmetry of
the universe. Specifically, we demonstrate that the viable corners of parameter
space may be identified by their proximity to an enhanced discrete symmetry
point. At this point, the global symmetry group of the theory is extended by a
discrete group under which the scalar sector is non-trivially charged, and the
discrete symmetry is spontaneously broken such that the discrete symmetry
relates degenerate electroweak preserving and breaking vacua. This idea is used
to investigate several specific models of the electroweak symmetry breaking
sector. The phase transitions identified through this method suggest
implications for other relics such as dark matter and gravitational waves.
