A normal subgroup is a subgroup that is invariant under conjugation by any element of the original group: H is normal if and only if g H g − 1 = H gHg^{-1} = H gHg−1=H for any. g \in G. Equivalently, a subgroup H of G is normal if and only if g H = H g gH = Hg gH=Hg for any g ∈ G g \in G g∈G. ...
How do you prove a subgroup is normal?
The best way to try proving that a subgroup is normal is to show that it satisfies one of the standard equivalent definitions of normality.
- Construct a homomorphism having it as kernel.
- Verify invariance under inner automorphisms.
- Determine its left and right cosets.
- Compute its commutator with the whole group.
What is it called normal subgroup?
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part.
