The hydrogen-bonded monohydroxyl alcohols form a large class of glass formers studied more than one hundred years, and still the structure and dynamics have continued to be a research problem. Recent advance suggests a hydrogen-bonded transient supramolecular structure, which is the origin of the Debye relaxation dominating the dielectric loss spectra of many monohydroxyl alcohols. Obscured by the slower Debye relaxation, the structural alpha-relaxation is either not resolved or showing up as a shoulder and the supposedly universal Johari-Goldstein (JG) beta-relaxation is not always observed. Thus, properties of the alpha-relaxation and the JG beta-relaxation as well as the strong connection between the two relaxations generally observed in other classes of glass formers are not commonly known in the monohydroxyl alcohols. Notwithstanding, extremely broadband dielectric relaxation and high-precision light scattering experiments published recently have resolved the alpha-relaxation and a secondary relaxation in two archetypal monohydroxyl alcohols, 1-propanol and 5-methyl-2-hexanol (5M2H) by Gabriel et al. We analyzed their experimental data and applied the Coupling Model to show that the secondary relaxations in 1-propanol and 5M2H are JG beta-relaxations with strong connection to the alpha-relaxation. The result is novel because it is not known before whether the secondary relaxations of these two monohydroxyl alcohols are JG beta-relaxation involving the entire molecule or are intramolecular relaxations. On the basis of this conclusion, we predict that the secondary relaxation is pressure dependent and the ratio tau(beta)(T,P)/tau(alpha)(T,P) is invariant to variations of P and T, whereas tau(alpha)(T,P) is maintained constant and provided that the frequency dispersion of the alpha-relaxation is also constant. The prediction is compared with the dielectric data of 5M2H at elevated pressures. On the basis of the identification of monohydroxyl alcohols as short-chain polymeric liquids by others, an explanation of the stronger T and P dependences of tau(alpha)(T,P) than the Debye relaxation time tau(D)(T,P) is given.