Frattini subgroup is nilpotent in finite
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Further information: Frattini subgroup
Finite nilpotent group
The result generalizes in two important respects. First, we can prove considerably stronger results about Frattini subgroups for finite groups, and more generally, for groups in which every proper subgroup is contained in a maximal subgroup. Second, we can generalize to proving the results about any Frattini-embedded normal subgroup of an arbitrary group.
- Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini
- Frattini-embedded normal-realizable implies ACIC
Given: A finite group , and is the Frattini subgroup
To prove: For any Sylow subgroup of , is normal in .
Proof: In fact, we shall show that is normal in .
Here's the idea. By applying Frattini's argument and the fact that , we have . Now if , it is contained in a maximal subgroup of . Since is contained in every maximal subgroup, , leading to a contradiction.
- Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Exercise 25, Page 199 (Section 6.2)