 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2013/paper-2/9/i/solution

N \leq G

g N g^{- 1} = N for all g \in G .

Equivalently,

g N = N g

for every

g

; normality permits the cosets to inherit a quotient-group multiplication.

ϕ : G_{1} \to G_{2}, ϕ (g h) = ϕ (g) ϕ (h) .

It consequently sends identity to identity and inverses to inverses, since

ϕ (e) = ϕ (e)^{2}

and

ϕ (g) ϕ (g^{- 1}) = ϕ (e)

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