normalizer (original) (raw)

Definitions

A subgroup H of G is said to be self-normalizing if NG⁢(H)=H.

Properties

NG⁢(H) is always a subgroup of G, as it is the stabilizer of H under the action (g,H)↦g⁢H⁢g-1of G on the set of all subsets of G(or on the set of all subgroups of G, if H is a subgroup).

If H is a subgroup of G, then H≤NG⁢(H).

If H is a subgroup of G, then H is a normal subgroup of NG⁢(H); in fact, NG⁢(H) is the largest subgroup of Gof which H is a normal subgroup. In particular, if H is a subgroup of G, then H is normal in G if and only if NG⁢(H)=G.