group theory

Question

Which of the following is not a group?

• Set of integer under multiplication operation.
• Set of natural number under multiplication.
• Set of natural number under addition.
• All of the above

Why is it not group under addition?

Ans given is d.

Comments

The Additive Inverse of any Natural Number is not a Natural Number.

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Okay...got it thanks :)

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The  Additive Identity ( i.e. 0) is not in the set itself.

Actually this reason should be preferred over the one I gave previously because we must first check for identity, if identity element is in the set then only we can talk about inverse of elements.

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I agree to your point if you take natural numbers to start from {1,2,3...} But there are places where natural numbers are considered from 0. In that case identity element would exist..inverse would still not exist.

Answer 1

Only option A is not the group. Since its not mentioned whether the group of integers are non zero or not as we will take the entire domain i.e all integers. Now every element of integer group satisfies group property except 0 where when    0*(1/0) = (1/0)*0 is not defined. Thus option A is not the group. 

Answer 2

None of them are group

• Set of integer under multiplication operation.
  • Reason: inverse doesn’t belong to integer set. inverse of 2 is ½.
• Set of natural number under multiplication.
  • Reason: inverse doesn’t belong to integer set. inverse of 2 is ½.
• Set of natural number under addition
  • Reason: identity element 0 doesn’t belong to natural number set.