Conjugate elements. Let 0,T E Sn. Define d = TOT-1. (o and d are called conjugate elements; see Chapter 6.) (a) Show that if o (i) = j, then 8(T(i)) = T(j). (b) Explain that the previous part says that if you apply T to each of the entries in the cycle notation of o, then you get the cycle notation for 8. In other words, if o has cycle decomposition (a1 a2 akı)(b1 b2 br2) ··· . ... ... 3. The Alternating Groups Then d has cycle decomposition (T(a1) T(a2) .. T(ak,))(T(b1) T(b2) .…. (c) Illustrate the previous part by letting o = (1 6 3)(7 5 2), and quickly writing down the cycle decomposition for TOT. Check your answer by actually finding the product TOT T(bk2)) -... (14 3 8)(2 6 5), т %3 тот-1.

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