10. Fill in the details of the proof that the symmetric groups S and So are...

 

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10. Fill in the details of the proof that the symmetric groups SÅ and So are isomorphic if |4| = || as follows: let 0: A → be a bijection. Define →SS by p(0)=00000-¹ for all o ESA P: So and prove the following: (a) p is well defined, that is, if o is a permutation of A then 0ooo0 is a permutation of N. (b) is a bijection from SA onto Sn. [Find a 2-sided inverse for p.] (c) p is a homomorphism, that is, 4(00T) = 4(o) o p(T). Note the similarity to the change of basis or similarity transformations for matrices (we shall see the connections between these later in the text). 10. Fill in the details of the proof that the symmetric groups SÅ and So are isomorphic if |4| = || as follows: let 0: A → be a bijection. Define →SS by p(0)=00000-¹ for all o ESA P: So and prove the following: (a) p is well defined, that is, if o is a permutation of A then 0ooo0 is a permutation of N. (b) is a bijection from SA onto Sn. [Find a 2-sided inverse for p.] (c) p is a homomorphism, that is, 4(00T) = 4(o) o p(T). Note the similarity to the change of basis or similarity transformations for matrices (we shall see the connections between these later in the text).

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