Is the subgroup of complex numbers with modules 1 a normal s

Is the subgroup of complex numbers with modules =1 a normal subgroup of the complex numbers without 0 under multiplication?

Solution

Yes.

Let G={z in C*: |z|=1}. Take any z from G and any w from C*=C\\{0}.

Then |wzw^-1| = |ww^-1z|= |z|= 1 as (C*, .) is abelian.

So wzw^-1 is in G for all z in G and w in C*.

i.e. G is normal in C*.