Frattini subgroup is finitely generated implies subset is generating set iff image in Frattini quotient is

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Statement with symbols

Suppose G is a group, and \Phi(G) (the Frattini subgroup of G) is a finitely generated group. Then, a subset S of G is a generating set for G if and only if the image of S in the Frattini quotient G/\Phi(G) under the quotient map, generates G/\Phi(G).