Find all conjugacy classes and their sizes in \(A_4\).
\(A_4\) is precisely the set of permutations with cycle type \((1,1,1,1)\), \((1,3)\) or \((2,2)\). We shall handle these case by case.
\((1,1,1,1)\): corresponds to the identity, the conjugacy class is simply \(\{1\}\).
\((1,3)\): there are \((4 \cdot 3 \cdot 2)/3 = 8\) possible 3-cycles. We will pick a representative \(x = (1\ 2\ 3)\). Note that \(C_{A_4}(x) = \left<x\right>\), since the only permutation fixing 1, 2 and 3 is the identity. The centraliser has order 3, so the conjugacy class containing \(x\) has \(12/3 = 4\) elements, meaning there are \(8/4 = 2\) conjugacy classes. Observe that for any 3-cycle \((a\ b\ c)\), there is some \(\tau \in S_4\) such that \(x^\tau = (\tau(1)\ \tau(2)\ \tau(3)) = (a\ b\ c)\). However, if we restrict \(\tau\) to be in \(A_4\) then \(x^\tau\) can only be in the conjugacy class of \(x\). We can infer that the 3-cycles in the other conjugacy class are conjugates of \(x\) by an odd permutation, which only swap 2 elements. Using this fact, we can list the other conjugacy class: \(\{(1\ 3\ 2), (4\ 2\ 3), (1\ 4\ 3), (1\ 2\ 4)\}\) (the first element swaps 2 and 3, while the rest swap 1, 2, and 3 with 4 respectively). Thus the conjugacy class of \(x\) consists of the remaining 3-cycles: \(\{(1\ 2\ 3), (4\ 3\ 2), (1\ 3\ 4), (1\ 4\ 2)\}\).
\((2,2)\): there are only 3 such permutations, and it can be noted that \(((1\ 2)(3\ 4))^{(2\ 3\ 4)} = (1\ 3)(2\ 4)\) and \(((1\ 2)(3\ 4))^{(2\ 4\ 3)} = (1\ 4)(2\ 3)\). Thus \(\{(1\ 2)(3\ 4), (1\ 3)(2\ 4), (1\ 4)(2\ 3)\}\) is a conjugacy class.
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