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When decomposing a finite semigroup into a wreath product of groups andaperiodic semigroups, complexity measures the minimal number of groups that areneeded. Determining an algorithm to compute complexity has been an open problemfor almost 60 years. The main result of this paper proves decidability ofKrohn-Rhodes complexity $c = 1$ of finite semigroups and automata. This isachieved by showing the lower bounds in work by Henckell, Rhodes and Steinbergfrom 2012 is sharp using profinite methods and results of McCammond from 1991and 2001.

Decidability of Krohn-Rhodes complexity $c = 1$ of finite semigroups and automata