d-Wise generation of prosolvable groups

Let G be a (topological) group. For 2⩽d∈N, denote by μd(G) the largest m for which there exists an m-tuple of elements of G such that any of its d entries generate G (topologically). We obtain a lower bound for μd(G) in the case when G is a prosolvable group. Our result implies in particular that if G is d-generated then the difference μd(G)−d tends to infinity when the smallest prime divisor of the order of G tends to infinity. One of the aim of the paper is to draw the attention to an intriguing question in linear algebra whose solution would allow to improve our bounds and determine the precise value for μd(G) in several relevant cases, for example when d=2 and G is a prosolvable group.