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Subgroup of $C^*$ (nonzero complex) with finite index.

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up vote 4 down vote favorite True or false: Let $C^*$ be the set of all nonzero complex numbers and $H$ be a subgroup of $C^*$(with respect to multiplication) be such that $[C^*:H]$ is finite then $H=C^*$. I'm guessing it true as I am thinking that if suppose there is such a proper subgroup $H$ for which the number of coset will be finite then I'm guessing that there is a gap between $C^*$ and $H$ and that gap cannot be filled up by finite union. But I am unable to give a concrete prove. Thanks in advance. abstract-algebra group-theory share | cite | improve this question asked Mar 20 '16 at 20:01 user322390 ...