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If $H$ is a subgroup of a group $G$, then $H$ is normal if any of the following equivalent conditions hold:

  1. $gHg^{-1}=H$ for all $g\in G$
  2. $gHg^{-1}\subseteq H$ for all $g\in G$
  3. $gH=Hg$ for all $g\in G$
  4. $(aH)*(bH)=(ab)H$ is a well-defined binary operation on the set of left cosets of $H$

If $H$ is a normal subgroup, we write $H \lhd G$, and the set of left cosets $G/H$ form a group under the operation given in (4) above.

Knowl status:
  • Review status: beta
  • Last edited by John Jones on 2019-05-23 19:00:12
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