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

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  • Group

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