- #1

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## Homework Statement

For ##S_3## compute the centralizers of each element and find the center.

## Homework Equations

## The Attempt at a Solution

I found that ##Z (S_3) = \{ 1 \}##, ##C_{S_3} (1) = S_3##, ##C_{S_3} ((13)) = \{1, (13) \}##, ##C_{S_3} ((12)) = \{1, (12) \}##, ##C_{S_3} ((23)) = \{1, (23) \}##, ##C_{S_3} ((123)) = \{1, (123) \}##, ##C_{S_3} ((132)) = \{ 1, (132)\}##.

However, I found these by direct computation of each pair of elements to see if they commute. Would there be a faster way to do this? Would Lagrange's theorem help reduce the number of calculations I needed to do?