Referring to Exercise 27, find all subgroups of S₃ (Example 8.7) that are conjugate to {P₀, µ₂}.

Data from Exercise 27

A subgroup H is conjugate to a subgroup K of a group G if there exists an inner automorphism i_g of G such that i_g[H] = K. Show that conjugacy is an equivalence relation on the collection of subgroups of G.

Data from Example 8.7.

An interesting example for us is the group S₃ of3! = 6 elements. Let the set A be {1, 2, 3}. We list the permutations of A and assign to each a subscripted Greek letter for a name.

The reasons for the choice of names will be clear later. Let

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(123). (231). 1 2 3 -(²9). 3 1 2 Po = P1 = P2 = = M1 = М2 = Из = 12 2 (133) 2 (1 2 3) 3 (273)