Cyclic Frattini quotient implies cyclic
Let be a group such that the following two conditions:
- The Frattini subgroup is a finitely generated group (note that this is automatically satisfied if is a finite group)
- The Frattini quotient, viz., the quotient by the Frattini subgroup, is a cyclic group
Then, is a cyclic group.
- Frattini subgroup is finitely generated implies subset is generating set iff image in Frattini quotient is