Show that if H is any group and is an element of with , then there is a unique homomorphism from to such that --> .

Please help. Thank you.

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- Sep 28th 2008, 09:19 PMdori1123homomorphism
Show that if H is any group and is an element of with , then there is a unique homomorphism from to such that --> .

Please help. Thank you. - Sep 30th 2008, 12:03 AMNonCommAlg
define by it's obvious that is a homomorphism and you only need to prove that is basically well-defined:

if then for some integer thus: hence is well-defined. this proves the*existence*of such homomorphism.

now suppose is another homomorphism with then for all hence which proves the*uniqueness*.