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Say I have a $\alpha: G \to H$.

  1. What do I have to do to prove that $\alpha$ is a homomorphism? I thought it was just show $\alpha(g_1g_2) = \alpha(g_1)\alpha(g_2)$. But someone told me today that I also have to show $\alpha(g^{-1}) = \alpha(g)^{-1}$. Is this the case and if so why?
  2. Why is $\alpha(g^{-1}) = \alpha(g)^{-1}$?
  • 3
    Homomorphism...of what? Groups, rings, algebras...?2012-11-22
  • 0
    You only need to show $\,\alpha(g_1\cdot g_2)=\alpha(g_1)*\alpha(g_2)\,$, if we're talking of groups $\,(G,\cdot)\,,\,(H,*)\,$2012-11-22
  • 0
    Set $g_2=e_G$ and you find $\alpha(e_G)=e_H$. Then set $g_2=g_1^{-1}$ and you have your implication.2012-11-22

5 Answers 5