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abstract: 'Gravitational instabilities play an important role in structure formation of gas-rich high-redshift disc galaxies. In this paper, we revisit the axisymmetric perturbation theory and the resulting growth of structure by taking the realistic thickness of the disc into account. In the unstable regime, which corresponds for thick discs to a Toomre parameter below the critical value $Q_{\mathrm{0,crit}} = 0.696$, we find a fastest growing perturbation wavelength that is always a factor 1.93 times larger than in the classical razor-thin disc approximation. This result is independent of the adopted disc scaleheight and by this independent of temperature and surface density. In order to test the analytical theory, we compare it with a high-resolution hydrodynamical simulation of an isothermal gravitationally unstable gas disc with the typical vertical ${\, \mathrm{sech} }^2$ density profile and study its break up into rings that subsequently fragment into dense clumps. In the first phase, rings form, that organize themselves discretely, with distances corresponding to the local fastest growing perturbation wavelength. We find that the disc scaleheight has to be resolved initially with five or more grid cells in order to guarantee proper growth of the ring structures, which follow the analytical prediction. These rings later 