Normal subgroup
A normal subgroup \(N\) of group \(G\) is one which is closed under conjugation: for all \(h \in G\), it is the case that \(\{ h n h^{-1} : n \in N \} = N\). In shorter form, \(hNh^{-1} = N\).
Since conjugacy is equivalent to “changing the worldview”, a normal subgroup is one which “looks the same from the point of view of every element of \(G\)”.
A subgroup of \(G\) is normal if and only if it is the kernel of some group homomorphism from \(G\) to some group \(H\). (Proof.)
why are they interesting
Children:
- Subgroup is normal if and only if it is the kernel of a homomorphism
The “correct way” to think about normal subgroups is as kernels of homomorphisms.
- Quotient by subgroup is well defined if and only if subgroup is normal
- Subgroup is normal if and only if it is a union of conjugacy classes
A useful way to think about normal subgroups, which meshes with their “closed under conjugation” interpretation.
Parents:
- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.