Creating a homomorphism with given generators

[Ultraworld]

I got the Dihedral Group D = <(1 2 3 4 5 6 ), (1 2)(3 4)(5 6)> and the symmetric group Sym(5).

Now I want to construct a homomorphism f : D --> Sym(5). Am I free to map the generators (1 2 3 4 5 6) and (1 2)(3 4)(5 6) to any element in Sym(5) as long holds:
f((1 2 3 4 5 6))⁶ = 1,
f((1 2)(3 4)(5 6))² = 1.

I tried
f((1 2 3 4 5 6)) = (1 2 3)(4 5),
f((1 2)(3 4)(5 6)) = (1 2).
Which seems to be fine but
f((1 2 3 4 5 6)) = (1 2 3)(4 5),
f((1 2)(3 4)(5 6)) = (1 4).
seems to fail?

Why?

 

[matt grime]

You can't just send them to *any* elements of the right order. Homomrphisms must preserve group structure. Like composition of elements.

 

[Ultraworld]

True but how do i know my first attempt is indeed a homomorphism and the 2nd one not. Now I "prooved" it by a computer program (MAGMA).

 

[matt grime]

You check if it is a homomorphism. You know what the definition of a homomorphism is, so check if the maps satisfy the definition.

 

[matt grime]

And remember that the dihedral group is defined by the relations, in this case, g^6=e, h^2=e and hgh=g^{-1}.