# Homomorphisms and subgroups

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I need a detailed explanation for this problem. Any help is greatly appreciated. Thanks!

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#### Solution Preview

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 ...

#### Solution Summary

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).

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