  We prove a refinement of the inequality by Hoffmann-Jorgensen that is
significant for three reasons. First, our result improves on the
state-of-the-art even for real-valued random variables. Second, the result
unifies several versions in the Banach space literature, including those by
Johnson and Schechtman [Ann. Probab. 17 (1989)], Klass and Nowicki [Ann.
Probab. 28 (2000)], and Hitczenko and Montgomery-Smith [Ann. Probab. 29
(2001)]. Finally, we show that the Hoffmann-Jorgensen inequality (including our
generalized version) holds not only in Banach spaces but more generally, in a
very primitive mathematical framework required to state the inequality: a
metric semigroup $\mathscr{G}$. This includes normed linear spaces as well as
all compact, discrete, or (connected) abelian Lie groups.
