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Prove the following theorem: Let φ: G→G′ be a group homomorphism, and let H=ker(φ). Let a∈G.Then...

Prove the following theorem: Let φ: G→G′ be a group homomorphism, and let H=ker(φ). Let a∈G.Then the set

(φ)^{-1}[{φ(a)}] ={x∈G|φ(x)} =φ(a)

is the left coset aH of H, and is also the right coset Ha of H. Consequently, the two partitions of G into left cosets and into right cosets of H are the same

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