Question

Which of the following statements is false?
(I) \(\mathbb{Z}_5\times\mathbb{Z}_7\) is a cyclic group.
(II) The order of the element \((2,5)\in\mathbb{Z}_5\times\mathbb{Z}_7\) is 10.
(III) \(\text{Aut}(\mathbb{Z}_5\times\mathbb{Z}_7)\) is isomorphic to \(\mathbb{Z}_{35}\).

• A. Only (I)
• B. Only (II)
• C. Only (II) and (III)
• D. Only (I) and (III)

Hint:

Since \(\gcd(5,7)=1\), use the Chinese Remainder Theorem for (I), the lcm rule for element order in (II), and \(\text{Aut}(\mathbb{Z}_m\times\mathbb{Z}_n)\cong\text{Aut}(\mathbb{Z}_m)\times\text{Aut}(\mathbb{Z}_n)\) for coprime \(m,n\) in (III).

Answer 1

The Correct Option is C

Step 1: Check statement (I).
Since \(\gcd(5,7)=1\), the Chinese Remainder Theorem gives \(\mathbb{Z}_5\times\mathbb{Z}_7 \cong \mathbb{Z}_{35}\), which is cyclic. Statement (I) is true.

Step 2: Check statement (II).
In \(\mathbb{Z}_5\), the order of \(2\) is \(5/\gcd(2,5)=5\). In \(\mathbb{Z}_7\), the order of \(5\) is \(7/\gcd(5,7)=7\). The order of \((2,5)\) in the direct product is\[\operatorname{lcm}(5,7) = 35\]not 10. Statement (II) is false.

Step 3: Check statement (III).
Since \(\mathbb{Z}_5\times\mathbb{Z}_7 \cong \mathbb{Z}_{35}\), we have \(\text{Aut}(\mathbb{Z}_5\times\mathbb{Z}_7) \cong \text{Aut}(\mathbb{Z}_{35}) \cong (\mathbb{Z}_{35})^{\times}\), of order \(\varphi(35)=\varphi(5)\varphi(7)=4\times 6=24\). A group of order 24 cannot be isomorphic to \(\mathbb{Z}_{35}\) (order 35). Statement (III) is false.

Step 4: Conclusion.
Statements (II) and (III) are false, while (I) is true.\[\boxed{\text{Only (II) and (III)}}\]