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 GL(n, ℝ) be the multiplicative group of invertible n × n matrices, and let ℝ be the additive group of real numbers. Let ϕ : GL(n, ℝ) → ℝ be given by ϕ (A) = tr(A), where tr(A) is defined in Exercise 13.
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