# 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

Children:

Parents:

• Group

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