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normal subgroup

A subgroup H \le G with gHg^{-1}=H for every g \in G -- equivalently, gH=Hg for every g.

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Examples

  • A_3 \trianglelefteq S_3: conjugating any element of A_3 by anything in S_3 lands back inside A_3.

  • Every subgroup of an abelian group, like \mathbb{Z}_6, is automatically normal.

See also

conjugate quotient group index of a subgroup
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