Abstract

Motivated by the well-known conjecture of Andrews and Curtis [Amer. Math. Monthly 73: 21–28, 1966.], we consider the question of how, in a given n-generator group G, any ordered n-tuple of “annihilators” of G, that is, with normal closure all of G, can be transformed by standard moves into a generating n-tuple. The recalcitrance of G is defined to be the least number of elementary standard moves (“elementary M-transformations”) by means of which every annihilating n-tuple can be transformed into a generating n-tuple. We obtain upper estimates for the recalcitrance of n-generator finite groups—thus quantifying a result from [Borovik, Lubotzky, and Myasnikov, Progr. Math. 248: 15–30, 2005]—and of a wide class of n-generator solvable groups, thus extending and correcting a result from [Burns, Herfort, Kam, Macedońska, and Zalesskii, Bull. Austral. Math. Soc. 60: 245–251, 1999].