In Exercises determine whether the given map ϕ is a homomorphism. [Hint: The straightforward way to proceed is to check whether ϕ (ab) = ϕ(a) ϕ(b) for all a and b in the domain of ϕ. However, if we should happen to notice that ϕ-1 [{e'}] is not a subgroup whose left and right cosets coincide, or that ϕ does not satisfy the properties given in Exercise 44 or 45 for finite groups, then we can say at once that ϕ is not a homomorphism.
Let Mn be the additive group of all n × n matrices with real entries, and let ℝ be the additive group of real numbers. Let ϕ (A) = det(A), the determinant of A, for A ∈ Mn.
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