The normalizer $N_G(H)=\{g\in G:gH{g}^{-1}=H\}$ of a subgroup

Depends on

• Subgroup
• Conjugation $x\mapsto gxg^{-1}$ is an automorphism

Used by

• A regular connected covering has deck group π₁(B,b₀)/p_*π₁(E,e₀) Corollary
• Frobenius automizer criterion for p nilpotence Corollary
• Cyclic sylow does not alone imply a normal p complement Counterexample
• Control of fusion in a sylow p subgroup Definition
• Frobenius complement and frobenius group Definition
• P local normalizer for normal complement theory Definition
• p-local subgroup Definition
• The three subgroups of order 2 in S₃ are conjugate and each is self-normalizing Example
• Abelian sylow fusion in its normalizer Lemma
• 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
• Frobenius kernel cardinality Lemma
• Fusion control and centralizer transitivity are equivalent Lemma
• Local normal p complements force control of fusion Lemma
• Local sylow conjugacy ascent for fusion Lemma
• Normal p complements pass to subgroups and p local normalizers Lemma
• P automizer condition implies fusion control Lemma
• Proper subgroup of a finite p group is properly normalized local Lemma
• The local automizer condition gives centralizer conjugacy of Sylow subgroups Lemma
• A subgroup containing the normalizer of a Sylow subgroup is self-normalizing Theorem
• Burnside normal p complement theorem 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)