Are the following mapings automorphisms of their respective groups

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Are the following mapings automorphisms of their respective groups

[From: ] [author: ] [Date: 11-11-07] [Hit: ]

T is not 1-1, and thus not an isomorphism.b) Since G is not abelian T(xy) = (xy)^(-1) is different from x^(-1) y^(-1) = T(x) T(y).Hence, T is not even a homomorphism.I hope this helps!......

a) G cyclic group of order 12, T:x -> x^3
b) G is the group S3, T: x -> x^-1

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a) Using multiplicative notation, let G = .
Note that ker T = {e, a^3, a^6, a^9}.

Hence, T is not 1-1, and thus not an isomorphism.

b) Since G is not abelian T(xy) = (xy)^(-1) is different from x^(-1) y^(-1) = T(x) T(y).
Hence, T is not even a homomorphism.

I hope this helps!

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