Question

Which one of the following statements is incorrect?
(I) The group \(\mathbb{Z}_{15}\) is isomorphic to \(\mathbb{Z}_3\times\mathbb{Z}_5\)
(II) \(\text{Inn}(A_3)\) is isomorphic to \(A_3\)

• A. Only (I)
• B. Only (II)
• C. Both (I) and (II)
• D. Neither (I) nor (II)

Hint:

Use the Chinese Remainder Theorem for statement (I), and recall \(\text{Inn}(G)\cong G/Z(G)\) for statement (II).

Answer 1

The Correct Option is B

Step 1: Check statement (I).
Since \(\gcd(3,5)=1\), the Chinese Remainder Theorem gives a group isomorphism\[\mathbb{Z}_{15} \cong \mathbb{Z}_3 \times \mathbb{Z}_5\]Both sides are cyclic of order 15, and the map \(x \bmod 15 \mapsto (x \bmod 3, x \bmod 5)\) is a well-defined bijective homomorphism. So statement (I) is correct, not incorrect.

Step 2: Check statement (II).
For any group \(G\), \(\text{Inn}(G) \cong G/Z(G)\), where \(Z(G)\) is the center. Here \(A_3 = \{e,(123),(132)\}\) is cyclic of order 3, and every abelian group equals its own center, so \(Z(A_3) = A_3\). Therefore\[\text{Inn}(A_3) \cong A_3/Z(A_3) = A_3/A_3 = \{e\}\]So \(\text{Inn}(A_3)\) is the trivial group of order 1, not isomorphic to \(A_3\) (order 3). Statement (II) is incorrect.

Step 3: Conclusion.
Only statement (II) is incorrect.\[\boxed{\text{Only (II)}}\]