Kernel of group homomorphism

The ker­nel of a group ho­mo­mor­phism \(f: G \to H\) is the col­lec­tion of all el­e­ments \(g\) in \(G\) such that \(f(g) = e_H\) the iden­tity of \(H\).

It is im­por­tant to note that the ker­nel of any group ho­mo­mor­phism \(G \to H\) is always a sub­group of \(G\). In­deed:

  • if \(f(g_1) = e_H\) and \(f(g_2) = e_H\) then \(e_H = f(g_1) f(g_2) = f(g_1 g_2)\), so the ker­nel is closed un­der \(G\)’s op­er­a­tion;

  • if \(f(x) = e_H\) then \(e_H = f(e_G) = f(x^{-1} x) = f(x^{-1}) f(x) = f(x^{-1})\) (where we have used that the image of the iden­tity is the iden­tity), so in­verses are con­tained in the pu­ta­tive sub­group;

  • \(f(e_G) = e_H\) be­cause the image of the iden­tity is the iden­tity, so the iden­tity is con­tained in the pu­ta­tive sub­group.

It turns out that the no­tion of “nor­mal sub­group” co­in­cides ex­actly with the no­tion of “ker­nel of ho­mo­mor­phism”. (Proof.) The “ker­nel of ho­mo­mor­phism” view­point of nor­mal sub­groups is much more strongly mo­ti­vated from the point of view of cat­e­gory the­ory; Ti­mothy Gow­ers con­sid­ers this to be the cor­rect way to in­tro­duce the teach­ing of nor­mal sub­groups in the first place.

Parents:

  • Group homomorphism

    A group ho­mo­mor­phism is a “func­tion be­tween groups” that “re­spects the group struc­ture”.