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We study the physical properties of a sample of 6 SMGs in the COSMOS field, spectroscopically confirmed to lie at z>4. We use new GMRT 325 MHz and 3 GHz JVLA data to probe the rest-frame 1.4 GHz emission at z=4, and to estimate the sizes of the star-forming (SF) regions of these sources, resp. Combining our size estimates with those available in the literature for AzTEC1 and AzTEC3 we infer a median radio-emitting size for our z>4 SMGs of (0.63"+/-0.12")x(0.35"+/-0.05") or 4.1x2.3 kpc^2 (major times minor axis; assuming z=4.5) or lower if we take the two marginally resolved SMGs as unresolved. This is consistent with the sizes of SF regions in lower-redshift SMGs, and local normal galaxies, yet higher than the sizes of SF regions of local ULIRGs. Our SMG sample consists of a fair mix of compact and more clumpy systems with multiple, perhaps merging, components. With an average formation time of ~280 Myr, derived through modeling of the UV-IR SEDs, the studied SMGs are young systems. The average stellar mass, dust temperature, and IR luminosity we derive are M*~1.4x10^11 M_sun, T_dust~43 K, and L_IR~1.3x10^13L_sun, resp. The average L_IR is up to an order of magnitude higher than for SMGs at lower redshifts. Our SMGs follow the correlation between dust temperature and IR luminosity as derived for Herschel-selected 0.1<z<2 galaxies. We study the IR-radio correlation for our sources and find a deviation from that derived for z<3 ULIRGs (<q_IR>=1.95+/-0.26 for our sample, compared to q~2.6 for IR luminous galaxies at z<2). In summary, we find that the physical properties derived for our z>4 SMGs put them at the high end of the L_IR-T_dust distribution of SMGs, and that our SMGs form a morphologically heterogeneous sample. Thus, further in-depth analyses of large, statistical samples of high-redshift SMGs are needed to fully understand their role in galaxy formation and evolution.