Question:
Which ONE of the following statements is FALSE?
Which ONE of the following statements is FALSE?
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When checking group homomorphisms, the kernel is important. A nontrivial kernel implies that some elements of the domain map to the identity in the codomain, which can often be verified through the properties of the groups involved.
Updated On: Jun 1, 2026
- There is a surjective group homomorphism from the additive group of rational numbers to the multiplicative group of all complex roots of unity.
- The multiplicative group of complex numbers of modulus one is isomorphic to a quotient group of the additive group of real numbers.
- Any group homomorphism from the multiplicative group of nonzero complex numbers into the group of all invertible \( 2 \times 2 \) matrices with real entries has a nontrivial kernel.
- There exists a group homomorphism from the symmetric group on \( n \) symbols into the multiplicative group of all invertible \( n \times n \) matrices with real entries, which has a trivial kernel.
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The Correct Option is C
Solution and Explanation
Step 1: Analyze statement (A).
This statement refers to a surjective homomorphism between two different types of groups, one additive (rationals) and the other multiplicative (complex roots of unity). This is indeed possible as the map exists and is surjective.
Step 2: Analyze statement (B).
The statement says that the multiplicative group of complex numbers of modulus one is isomorphic to a quotient group of the additive group of real numbers. This is correct, as the quotient group structure can be seen to align with the multiplicative nature of complex numbers on the unit circle.
Step 3: Analyze statement (C).
This is the false statement. The kernel of a homomorphism from the multiplicative group of nonzero complex numbers into \( 2 \times 2 \) invertible matrices is trivial because a nontrivial kernel would imply nontrivial relations between matrices that are not present in this case. Therefore, this statement is false.
Step 4: Analyze statement (D).
This statement refers to a homomorphism from the symmetric group on \( n \) symbols into the multiplicative group of invertible \( n \times n \) matrices. Such a homomorphism exists and has a trivial kernel. This statement is true.
Step 5: Conclusion.
From the analysis, statement (C) is the false statement, as it wrongly suggests a nontrivial kernel for a homomorphism that should have a trivial kernel. Thus, the correct answer is (C).
This statement refers to a surjective homomorphism between two different types of groups, one additive (rationals) and the other multiplicative (complex roots of unity). This is indeed possible as the map exists and is surjective.
Step 2: Analyze statement (B).
The statement says that the multiplicative group of complex numbers of modulus one is isomorphic to a quotient group of the additive group of real numbers. This is correct, as the quotient group structure can be seen to align with the multiplicative nature of complex numbers on the unit circle.
Step 3: Analyze statement (C).
This is the false statement. The kernel of a homomorphism from the multiplicative group of nonzero complex numbers into \( 2 \times 2 \) invertible matrices is trivial because a nontrivial kernel would imply nontrivial relations between matrices that are not present in this case. Therefore, this statement is false.
Step 4: Analyze statement (D).
This statement refers to a homomorphism from the symmetric group on \( n \) symbols into the multiplicative group of invertible \( n \times n \) matrices. Such a homomorphism exists and has a trivial kernel. This statement is true.
Step 5: Conclusion.
From the analysis, statement (C) is the false statement, as it wrongly suggests a nontrivial kernel for a homomorphism that should have a trivial kernel. Thus, the correct answer is (C).
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