Alternating Group - Conjugacy Classes

Conjugacy Classes

As in the symmetric group, the conjugacy classes in An consist of elements with the same cycle shape. However, if the cycle shape consists only of cycles of odd length with no two cycles the same length, where cycles of length one are included in the cycle type, then there are exactly two conjugacy classes for this cycle shape (Scott 1987, §11.1, p299).

Examples:

  • the two permutations (123) and (132) are not conjugates in A3, although they have the same cycle shape, and are therefore conjugate in S3
  • the permutation (123)(45678) is not conjugate to its inverse (132)(48765) in A8, although the two permutations have the same cycle shape, so they are conjugate in S8.

Read more about this topic:  Alternating Group

Famous quotes containing the word classes:

    “My plan of instruction is extremely simple and limited. They learn, on week-days, such coarse works as may fit them for servants. I allow of no writing for the poor. My object is not to make fanatics, but to train up the lower classes in habits of industry and piety.”
    —Hannah More (1745–1833)