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Calculus and Beyond Homework Help
Do Isomorphic Groups Have Isomorphic Centers?
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[QUOTE="Mr Davis 97, post: 6050808, member: 515461"] [h2]Homework Statement [/h2] Prove that if the groups ##G \cong H## are isomorphic then ##Z(G) \cong Z(H)## [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] Let ##\phi: G \to H## be an isomorphism. Define ##f: Z(G) \to Z(H)## s.t ##f(z) = \phi (z)##. First, we will show that this map is well-defined, in the sense that elements in ##Z(G)## are always mapped to elements in ##Z(H)## by ##f##: Let ##z \in Z(G)##. Then ##zg=gz## for all ##g \in G##. This implies that ##f(zg)=f(gz) \implies \phi (z) \phi (g) = \phi (g) \phi (z)## for all ##g##. But ##g## is arbitrary and ##\phi## is bijective, so ##\phi (g)## is an arbitrary element of ##H##. Hence we have ##\phi(z)h=h \phi(z)## for all ##h \in H##, which means ##f(z)h=h f(z)## for all ##h \in H##. So ##f(z) \in Z(H)##. Next, we show that ##f## is an isomorphism: It is clear that ##f## is a homomorphism, since it is defined in terms of ##\phi##. It remains to show that ##f## is a bijection: ##f## is invertible, and the inverse is ##f^{-1} = \phi^{-1}##. Hence ##Z(G) \cong Z(H)## [/QUOTE]
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Do Isomorphic Groups Have Isomorphic Centers?
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