Cyclic Frattini quotient implies cyclic

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Statement

Let G be a group such that the following two conditions:

  1. The Frattini subgroup \Phi(G) is a finitely generated group (note that this is automatically satisfied if G is a finite group)
  2. The Frattini quotient, viz., the quotient by the Frattini subgroup, is a cyclic group

Then, G is a cyclic group.

Facts used

Proof

The proof is more or less direct from the above stated fact. PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

References