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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:Lenticular galaxies
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 as far 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 from simple 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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