Cyclic Frattini quotient implies cyclic
From Groupprops
Contents
Statement
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.
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
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 173, (Section 5.1, Theorem 1.2)