Generalized bases of finite groups

Motivated by recent results on the minimal base of a permutation group, we introduce new local invariant attached to arbitrary finite groups. More precisely, subset Delta group G is called p-base (where p prime) if generates p-group and C_G(Delta) p-nilpotent. Building Halasi-Mar\'oti, prove that p-solvable groups possess p-bases size 3 for every prime p. For other prominent exhibit 2. In fact, conjecture existence 2 group. Finally, notion generalized blocks fusion systems.