Image of the identity under a group homomorphism is the identity

For any group ho­mo­mor­phism \(f: G \to H\), we have \(f(e_G) = e_H\) where \(e_G\) is the iden­tity of \(G\) and \(e_H\) the iden­tity of \(H\).

In­deed, \(f(e_G) f(e_G) = f(e_G e_G) = f(e_G)\), so pre­mul­ti­ply­ing by \(f(e_G)^{-1}\) we ob­tain \(f(e_G) = e_H\).

Parents:

  • Group homomorphism

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