Cyclic Frattini quotient implies cyclic

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

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

Proof

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

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References

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 173, (Section 5.1, Theorem 1.2)