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.)

knows-requisite(category theory equaliser): From a category-theoretic point of view, the kernel of \(f\) is an equaliser of an arrow \(f\) with the zero arrow; it is therefore universal such that composition with \(f\) yields zero.

why are they interesting



  • Group

    The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.