Recalcitrance in groups

Abstract

Motivated by a well-known conjecture of Andrews and Curtis, we consider the question as to how in a given n-generator group G, a given set of n “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 of G can be transformed into a generating n-tuple. We show that in the classes of finite and soluble groups, having zero recalcitrance is equivalent to nilpotence, and that a large class of 2-generator soluble groups has recalcitrance at most 3. Some examples and remarks are included.

Recalcitrance in groups II

“…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 [2]-and of a wide class of n-generator solvable groups, thus extending and correcting a result from [3]. …”

“…This paper represents a continuation-and a correction of one result-of the investigation begun in [3]. Thus again the question of interest here is whether, and if so then with what degree of di‰culty, a given annihilating n-tuple ðr 1 ; .…”

“…We call such a transformation an M-transformation (M for ''modulo'' since s i is replaced by any element congruent to it modulo the other s j ). For some purposes-see [3,4]-the latter transformations are more convenient than those of Andrews and Curtis, and it is these that we shall take as our ''elementary moves''. In particular, they would seem to lend themselves more readily than elementary AC-transformations to quantifying the di‰culty of transforming a given annihilating n-tuple of a group into a generating n-tuple, since in general a single M-transformation is equivalent to several successive elementary AC-transformations.…”

“…In particular, they would seem to lend themselves more readily than elementary AC-transformations to quantifying the di‰culty of transforming a given annihilating n-tuple of a group into a generating n-tuple, since in general a single M-transformation is equivalent to several successive elementary AC-transformations. As in [3], we define the recalcitrance of an annihilating n-tuple of a group G of rank n to be the least number of M-transformations needed to transform that n-tuple into a generating n-tuple for G. We then define the recalcitrance of the group G (recðGÞ) to be the supremum of the recalcitrances of all of its annihilating n-tuples.…”

“…Although in [3] the n-tuples were unordered, it seems preferable to use ordered n-tuples since, as noted in [3,Remark 4], the least number of annihilating elements needed by the group then becomes unimportant.…”