- #1
Ultraworld
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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?
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?