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Let H and K be finite abelian groups and let G be an infinite abelia.pdf

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Let H and K be finite abelian groups and let G be an infinite abelian group. Let it be known that G x H is isomorphic to G x K . Show by example that because G is infinite then H need not be isomorphic to K. Solution Use the fundamental theorem for finitely generated abelian groups. First show that the finite parts of G and H are equal, then show that the infinite parts are equal. For example if G,H,K have infinite parts Z^n, Z^m and Z^ p, then Z^(n+p)=Z^(m+p) implies n=m.

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Let H and K be finite abelian groups and let G be an infinite abelian group. Let it be known that G x H is isomorphic to G x K . Show by example that because G is infinite then H need not be isomorphic to K. Solution Use the fundamental theorem for finitely generated abelian groups. First show that the finite parts of G and H are equal, then show that the infinite parts are equal. For example if G,H,K have infinite parts Z^n, Z^m and Z^ p, then Z^(n+p)=Z^(m+p) implies n=m

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Let H and K be finite abelian groups and let G be an infinite abelia.pdf

  • 1. Let H and K be finite abelian groups and let G be an infinite abelian group. Let it be known that G x H is isomorphic to G x K . Show by example that because G is infinite then H need not be isomorphic to K. Solution Use the fundamental theorem for finitely generated abelian groups. First show that the finite parts of G and H are equal, then show that the infinite parts are equal. For example if G,H,K have infinite parts Z^n, Z^m and Z^ p, then Z^(n+p)=Z^(m+p) implies n=m