Anonymous asked in Science & MathematicsMathematics · 1 month ago

Isomorphism question?

Please see attached image.

Is this map an isomorphism?

I understand that I have to show its bijective and a homomorphism but unsure whether these exist for the function. I would appreciate any help. Thanks

Attachment image

1 Answer

  • David
    Lv 4
    1 month ago
    Favorite Answer

    To show (or not) it is bijective, you need to show it is injective and surjective.

    Pick any two elements of G, and show that θ(A) = θ(B) implies that A = B. This proves θ is injective. Since θ is just the inverse, this is quite simple (just utilise the definition of inverse, AA⁻¹ = I)

    To show it is surjective, pick an element A in the range (which is just G again). Then just find a B in G such that θ(B) = A. This is simple again, using the definition of inverse.

    Alternatively, you could just show that the inverse map exists. A map F is bijective if and only if it has an inverse. Since θ is just the inverse, the map is in fact self-inverse.

    You now want to show (or not) that θ is a homomorphism. To do this, utilise the definition of a homomorphism, θ(A•B) = θ(A)•θ(B) for all A, B in G.

    Then just plug in the definition of θ and see what happens.

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