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The Frattini subgroup as the intersection of the maximal subgroups of a finite group
Definition
For a finite group , the Frattini subgroup is If , the family is empty and its intersection inside is itself. Thus .
Depends on
Used by
- An extraspecial p-group of order p¹⁺²ⁿ has generator rank 2n Corollary
- Every element outside Φ(P) belongs to a minimal generating set of P Corollary
- Generation of a finite group is detected modulo its Frattini subgroup Corollary
- Maximal subgroups of a finite p-group are the inverse images of Frattini hyperplanes Corollary
- A nonsurjective homomorphism need not carry the Frattini subgroup into the target Frattini subgroup Counterexample
- Special and extraspecial p-groups Definition
- The Fitting and Frattini subgroups of S₃ Example
- The p-cores, Fitting subgroup, and Frattini subgroup of S₄ Example
- FALSE: every homomorphism carries the Frattini subgroup into the target Frattini subgroup False statement
- FALSE: the Frattini subgroup is the union of the maximal subgroups False statement
- The Frattini subgroup consists exactly of the nongenerators of a finite group Lemma
- The Frattini subgroup of a finite group is characteristic Lemma
- An automorphism of an extraspecial p-group acting trivially on its Frattini quotient is inner Proposition
- Three equivalent descriptions of an extraspecial p-group Proposition
- Nilpotence lifts over the Frattini subgroup of a finite group Theorem
- Sylow and maximal-subgroup characterizations of finite nilpotence Theorem
- The Frattini quotient is the largest elementary abelian quotient of a finite p-group Theorem
Dependency tree · two levels
2 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- David A. Craven, Finite Group Theory, Sections 1.4 and 2.3 (standard reference, not scraped)