  In this paper we address the question of synchronizing random automata in the
critical settings of almost-group automata. Group automata are automata where
all letters act as permutations on the set of states, and they are not
synchronizing (unless they have one state). In almost-group automata, one of
the letters acts as a permutation on $n-1$ states, and the others as
permutations. We prove that this small change is enough for automata to become
synchronizing with high probability. More precisely, we establish that the
probability that a strongly connected almost-group automaton is not
synchronizing is $\frac{2^{k-1}-1}{n^{2(k-1)}}(1+o(1))$, for a $k$-letter
alphabet.
