  The Khinchin-Kahane inequality is a fundamental result in the probability
literature, with the most general version to date holding in Banach spaces.
Motivated by modern settings and applications, we generalize this inequality to
arbitrary metric groups which are abelian.
  If instead of abelian one assumes the group's metric to be a norm (i.e.,
$\mathbb{Z}_{>0}$-homogeneous), then we explain how the inequality improves to
the same one as in Banach spaces. This occurs via a "transfer principle" that
helps carry over questions involving normed metric groups and abelian normed
semigroups into the Banach space framework. This principle also extends the
notion of the expectation to random variables with values in arbitrary abelian
normed metric semigroups $\mathscr{G}$. We provide additional applications,
including studying weakly $\ell_p$ $\mathscr{G}$-valued sequences and related
Rademacher series.
  On a related note, we also formulate a "general" Levy inequality, with two
features: (i) It subsumes several known variants in the Banach space
literature; and (ii) We show the inequality in the minimal framework required
to state it: abelian metric groups.
