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I'm doing an exercise from my textbook of abstract algebra class. And the book answer says that $(\Bbb R,\cdot)$ is not a subgroup of nonzero complex numbers under multiplication. But I thought:

  1. The set of $\Bbb R$ does have $1$, which is the identity of nonzero complex numbers under multiplication.
  2. $\Bbb R$ is closed under multiplication.
  3. $\Bbb R$ does have inverse for each of its member.

Any hints or pointing out my mistakes are very appreciated!

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    $\begingroup$ What is the inverse of $0$? $\endgroup$ Feb 11, 2018 at 22:34

1 Answer 1

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Since $\mathbb R$ isn't even a subset of $\mathbb{C}\setminus\{0\}$, it cannot be a subgroup.

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