Question

Let \( H \) be the subset of \( S_3 \) consisting of all \( \sigma \in S_3 \) such that \[ {Trace}(A_1 A_2 A_3) = {Trace}((A_1 \sigma(A_2) A_3)), \] for all \( A_1, A_2, A_3 \in M_2(\mathbb{C}) \). The number of elements in \( H \) is equal to ……… (answer in integer).

Hint:

For problems involving symmetric groups, analyze the structure of permutations and their properties.

Answer 1

Step 1: Symmetric group \( S_3 \). The group \( S_3 \) consists of all permutations of three elements. For \( \sigma \in H \), the trace property implies that \( \sigma \) must preserve the cyclic order of multiplication. 
Step 2: Identifying elements of \( H \). The elements of \( H \) are the identity \( e \), \( (1 \, 2 \, 3) \), and \( (1 \, 3 \, 2) \), as these permutations preserve the cyclic order. 
Step 3: Counting elements. Thus, \( |H| = 3 \). 
Step 4: Conclusion. The number of elements in \( H \) is \( {3} \).