Which of the following statements is false?
1996
Which of the following statements is false?
- A.
The set of rational numbers is an abelian group under addition
- B.
The set of integers is an abelian group under addition
- C.
The set of rational numbers forms an abelian group under multiplication
- D.
The set of real numbers excluding zero is an abelian group under multiplication
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Correct answer: C
The set of rational numbers is an abelian group under addition, and the set of integers is also an abelian group under addition. The non-zero real numbers form an abelian group under multiplication.
However, the set of all rational numbers under multiplication is not a group, because 0 is a rational number and 0 has no multiplicative inverse. Since one element lacks an inverse, the group axiom fails.
Therefore, option C is the false statement.