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Verify whether the following mapping is a homomorphism. If so, determine its kernel.

$f(x)=x^3$, for all $x$ belonging to $G$.

phi(a*b) = phi(a) * phi(b)

f(x*y) = (x*y)^3 = x^3 * y^3 = f(x) * f(y)

thus f is a homomorphism.

kernel of phi is the mapping of the identity element of the domain, in this case the multiplicative identity which is 1 considering x to be in the set of R – {0} i.e R+.

if that is the case then kernel of f is f(1) which is 1.
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