Normal subgroup

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The article on this topic in the Group Properties Wiki can be found at: normal subgroupTemplate:Subgroup property

Definition

A subgroup H of a group G is termed normal if it satisfies the following equivalent conditions:

  • H is the kernel of a homomorphism from G, i.e. there is a homomorphism \phi:G \to K of groups such that \phi^{-1}(e) = H
  • xHx^{-1} \subseteq H, or in other words, xhx^{-1} \in H for all x \in G, h \in H
  • xHx^{-1} = H

Facts

Normal subgroup and quotient goup

Normal subgroups of the fundamental group

Normal subgroups of the structure group