# isomorphisms external direct product

• Oct 19th 2010, 02:05 PM
stumped765
isomorphisms external direct product
Suppose we have some isomorphism φ which maps Z(subscript 3) + Z(subscript 5)
(ie the external direct product between Z3 and Z5..sorry dont know how to make the symbols) to Z(subscript 15)

If φ(2,3) = 2 which element in the external direct product maps to 1 in Z15

This is screaming chinese remainder thm to me but it doesnt work. I thought for sure the mapping would be something along the lines of 2*3 + 3*5 = ___ (mod 15)
but clearly doesnt work for the given example. ive tried numerous other different combinations of the same type of idea I just feel like im playing a "guess and check" type of game, any thoughts on a method?
• Oct 19th 2010, 07:22 PM
tonio
Quote:

Originally Posted by stumped765
Suppose we have some isomorphism φ which maps Z(subscript 3) + Z(subscript 5)
(ie the external direct product between Z3 and Z5..sorry dont know how to make the symbols) to Z(subscript 15)

If φ(2,3) = 2 which element in the external direct product maps to 1 in Z15

This is screaming chinese remainder thm to me but it doesnt work. I thought for sure the mapping would be something along the lines of 2*3 + 3*5 = ___ (mod 15)
but clearly doesnt work for the given example. ive tried numerous other different combinations of the same type of idea I just feel like im playing a "guess and check" type of game, any thoughts on a method?

As $2^4=1\!\!\pmod{15}$ , we get that $1=2^4=\phi(2,3)^4=\phi(1,1)$ ...which is hardly surprising since

a homom. of groups (or unitary rings) must map the unit (the multiplicative unit) to the unit...

Tonio