LetH= {expim*theta; theta in R}. Show thatthe map G -> H, exp(i*theta) |-> exp(im*theta) is a group homomorphismi if and only if m is an integer.(adsbygoogle = window.adsbygoogle || []).push({});

(=>) Assume G - > is a group homo s.t. phi(exp(I*theta)) = (exp(I*m*theta))

Consider exp(ix), exp(iy) in G

Since G is a homo,

phi(exp(ix)exp(iy)) = phi(exp(ix))phi(exp(iy)))

phi(exp(i(x+y))) = phi(exp(ix))phi(exp(iy)))

exp(im(x+y) =exp(imx)exp(imy)

Isthe m on the left equal to the m on the right? I don't think so, what's wrong here?

How do I show the converse?

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# Show something's a homomorphism

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