# Creating a homomorphism with given generators

1. Apr 18, 2007

### 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?

2. Apr 18, 2007

### matt grime

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

3. Apr 18, 2007

### 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).

4. Apr 18, 2007

### 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.

5. Apr 18, 2007

### 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}.