Normal subgroup

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Definition

Let (G,×) be a group and H a subgroup of G, we say H is a normal subgroup[1] of G if:

  • \forall x\in G[xH=Hx] where the xH and Hx are left and right cosets
    • This is the sameas saying: \forall x\in G[xHx^{-1}=H]

According to Serge Lang[1] this is equivalent (that is say if and only if or \iff)

  • H is the kerel of some homomorphism of G into some other group

Proof of claims

[Expand]

Claim 1: \forall x\in G[xH=Hx]\iff\forall x\in G[xHx^{-1}=H]


References

  1. Jump up to: 1.0 1.1 Undergraduate Algebra - Serge Lang