Homomorphisms and subgroups
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Please see the attachment.
(a) This non-trivial group homomorphism does not exist.
Proof: Suppose is a non-trivial group homomorphism. We consider an element in . We know is the set of all even permutations and each permutation can be expression as the multiplication of disjoint cycles. So can only be the ...
This shows how to give examples of a non-trivial group homomorphism, a basis for a given homomorphism, and 2-Sylow subgroups (or explain why none exists).