Normalizer-relatively normal subgroup

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This article describes a property that can be evaluated for a triple of a group, a subgroup of the group, and a subgroup of that subgroup.
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Suppose H \le K \le G are groups. We say that H is a normalizer-relatively normal subgroup of K with respect to G if the following equivalent conditions are satisfied:

  • Whenever K \le L \le G is such that K is normal in L, H is also normal in L.
  • The normalizer N_G(K) is contained in the normalizer N_G(H).
  • H is normal in N_G(K).

Relation with other properties

Stronger properties

Weaker properties