Alphabeta MathHow statement and proof provenance work
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The normalizer of a subgroup
Definition
Let be a subgroup (Subgroup). The normalizer of in is
Thus exactly when the conjugation automorphism preserves setwise (Conjugation is an automorphism). The subgroup property is proved in and are subgroups of ↗.
Depends on
Used by
- A regular connected covering has deck group π₁(B,b₀)/p_*π₁(E,e₀) Corollary
- p-local subgroup Definition
- The three subgroups of order 2 in S₃ are conjugate and each is self-normalizing Example
- C_G(x) and N_G(H) are subgroups of G Lemma
- Deck transformations of a connected covering correspond to cosets in the subgroup normalizer Lemma
- A subgroup containing the normalizer of a Sylow subgroup is self-normalizing Theorem
- Deck(E/B)≅ N_G(H)/H for a connected covering Theorem
- Every proper subgroup of a finite nilpotent group is properly contained in its normalizer Theorem
- Frattini argument: if N is normal in G and P is Sylow in N, then G=N N_G(P) Theorem
- Sylow II: in a finite group every p-subgroup lies in a conjugate of any Sylow p-subgroup, and the Sylow p-subgroups form a single conjugacy class Theorem
- The conjugates of a proper subgroup do not cover a finite group Theorem
- The conjugates of H are in bijection with G/N_G(H) and, for finite G, number [G:N_G(H)] Theorem
Dependency tree · two levels
8 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
- P. Brosnan, Undergraduate Algebra Notes, 3.14: G-Sets (standard reference, not scraped)