Normal Subgroup - Properties

Properties

  • Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.
  • Normality is preserved on taking direct products
  • A normal subgroup of a normal subgroup of a group need not be normal in the group. That is, normality is not a transitive relation. However, a characteristic subgroup of a normal subgroup is normal. Also, a normal subgroup of a central factor is normal. In particular, a normal subgroup of a direct factor is normal.
  • Every subgroup of index 2 is normal. More generally, a subgroup H of finite index n in G contains a subgroup K normal in G and of index dividing n! called the normal core. In particular, if p is the smallest prime dividing the order of G, then every subgroup of index p is normal.

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