Compute the indicated quantities for the given homomorphism¢.

Ker (∅) for ∅ : S₃ →Z₂ in Example 13.3

Data from Example 13.3

Let S_n be the symmetric group on n letters, and let ∅: S_n → Z₂ be defined by

Show that ∅ is a homomorphism.

Solution: We must show that ∅(σµ) = ∅(σ) + ∅(µ)for all choices of σ,µ ∈ S_n. Note that the operation on the right-hand side of this equation is written additively since it takes place in the group Z₂. Verifying this equation amounts to checking just four cases: 

σ odd and µ odd, 

σ odd and µ even, 

σ even and µ odd, 

σ even and µ even.

Data  from Exercise 46

Let a group G be generated by { a_i | i ∈ I}, where I is some indexing set and a_i ∈ G for all i ∈ I. Let ∅ : G → G' and µ : G → G' be two homomorphisms from G into a group G', such that ∅(a_i) = µ(a_i) for every i ∈ I. Prove that ∅ = µ. [Thus, for example, a homomorphism of a cyclic group is completely determined by its value on a generator of the group.]

() = = 0 1 ifo is an even permutation, if o is an odd permutation.