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Category:United States–Iran relationsNGC 434 NGC 434 is a lenticular galaxy located around 41 million light-years away in the constellation of Coma Berenices. It was discovered by astronomer Heinrich Louis d'Arrest on September 24, 1864. NGC 434 is a member of the NGC 4761 Group. See also List of NGC objects (4001–5000) References External links 434 41060 70222 Category:Lenticulargalaxies Category:Astronomical objects discovered in 1864 Category:Coma Berenices (constellation) Category:Discoveries by Heinrich Louis d'ArrestQ: Limits for Corollary 6.3 of A First Course in Abstract Algebra by Herstein I'm doing my homework for an abstract algebra course. My assignment is about using the category of abelian groups to prove Corollary 6.3 of Herstein's book. I have got asfar as the following theorem Theorem: Let $I$ be an index set. Then there is a surjective morphism $$\phi:G(I) \to A^I$$ where $G(I)=\{g:\mathbf{Z}^+ \to A|g(i)=g(i+1)\,\,\,\forall i \in I\}$ This allows me to conclude Corollary 6.3: Let $A$ be an abelian group and let $I$ be an index set. Then the collection $A^I$ of functions $\mathbf{Z}^+ \to A$ is countable. I have already proven this fromsimple cardinal arithmetic. The question is, is there some way of doing this using the categorical definition that Herstein provides in the text? I first tried a mapping $h:G(I) \to 2^I$ as follows. Let $g:I \to 2$ be given. Then $$h(g):2^I \to 2^2 \mapsto \{(f,g) \in 2^2|f(i)=g(i) \,\, \forall i \in I\}$$ I haven't yet proved this map is surjective, but it should be easy to prove

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