Say we have a homorphism $\phi : G \longrightarrow H$. I'm just trying to understand how will the elements of the center of $G$ behave under $\phi$ especially if $ \phi(G)=H$.
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1Elements of $Z(G)$ will map to elements of $Z(\phi(G))$ but not necessarily to elements of $Z(H)$. – Derek Holt Dec 31 '13 at 18:51
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3Um, you mean that $\phi(G) = H$? And unless we have a specific question, all we can give you is that $\phi(Z(G)) \subseteq Z(\phi(G))$. – Patrick Da Silva Dec 31 '13 at 18:52
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Let $z $ be in the center of $G$. Then for any $g\in G$ we have the following:
$\phi(z)\phi(g) = \phi(zg) = \phi(gz) = \phi(g)\phi(z)$. So $\phi$ maps an element of the center of $G$ to an element that commutes with the image of $G$ under $\phi$. Thus if $\phi(G)= H$ then $\phi(Z(G)) \subseteq Z(H)$.
Vladhagen
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