Question

The number of group homomorphisms from the group \(\mathbb{Z}_4\) to the group \(S_3\) is _________.

Hint:

For a cyclic domain group \(\mathbb{Z}_n\), each homomorphism is determined by an element of the codomain whose order divides \(n.\)

Answer 1

Correct Answer: 4

Step 1: Key property of homomorphisms.
A homomorphism from \(\mathbb{Z}_4 = \langle a \rangle\) to \(S_3\) is determined by the image of the generator \(a\). The element \(f(a)\) must satisfy \[ f(a)^4 = e, \] where \(e\) is the identity in \(S_3\).
Step 2: Possible images.
We need elements in \(S_3\) whose order divides 4. In \(S_3\), elements have orders \(1, 2, 3.\) Hence, only elements of order \(1\) or \(2\) can be chosen.
Step 3: Counting such elements.
- 1 element of order 1 (the identity). - 3 transpositions of order 2. Total \(= 4.\)
Step 4: Conclusion.
Hence, there are \(\boxed{4}\) group homomorphisms.