Is it possible that an element of a sylow basis of a finite solvable group G lies in the frattini subgroup of G?(i.e could an element of a sylow basis of a finite solvable group G be a non-generator subgroup?) 



sylow basis of finite solvable groups
You should tell the reader what a Sylow basis is. Assuming that it s a set of pairwise permutable Sylow subgroups of $G,$ one for each prime divisor, the answer is no. For if $P\_{1}$ is a Sylow $p\_{1}$-subgroup of $G$ belonging to a basis, then deleting $P\_{1}$from the generating set given by the Sylow basis leaves a set which generates a Hall $p\_{1}^{\prime}$-subgroup of $G$- in particular, a proper subgroup of $G.$


