We introduce the Gaussian part of a compact quantum group $\mathbb{ G}$, namely the largest quantum subgroup of $\mathbb{ G}$ supporting all the Gaussian functionals of $\mathbb{ G}$. We prove that the Gaussian part is always contained in the Kac part, and characterise Gaussian parts of classical compact groups, duals of classical discrete groups and $q$-deformations of compact Lie groups. The notion turns out to be related to a new concept of "strong connectedness" and we exhibit several examples of both strongly connected and totally strongly disconnected compact quantum groups.
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