2-hypernormalized satisfies intermediate subgroup condition

From Groupprops
Jump to: navigation, search
This article gives the statement, and possibly proof, of a subgroup property (i.e., 2-hypernormalized subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about 2-hypernormalized subgroup |Get facts that use property satisfaction of 2-hypernormalized subgroup | Get facts that use property satisfaction of 2-hypernormalized subgroup|Get more facts about intermediate subgroup condition


Statement

Suppose H is a 2-hypernormalized subgroup of a group G. In other words, N_G(N_G(H)) = G. Suppose K is a subgroup of G containing H. Then, H is 2-hypernormalized in K: N_K(N_K(H)) = K.

Related facts

Facts used

  1. Normality satisfies transfer condition

Proof

Given: H \le K \le G, N_G(N_G(H)) = G.

To prove: N_K(N_K(H)) = K.

Proof:

  1. N_K(H) = N_G(H) \cap K: This follows directly from the definition of normalizer.
  2. N_K(H) is normal in K: By assumption, N_G(H) is normal in G. Fact (1) yields that N_G(H) \cap K = N_K(H) is normal in K.
  3. N_K(N_K(H)) = K: This follows directly from the previous step.