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For instance, the alternating group of degree 4 is A''4'' = {e, (123), (132), (124), (142), (134), (143), (234), (243), (12)(34), (13)(24), (14)(23)}. BASIC PROPERTIES For ''n'' > 1, the group A''n'' is a . The group ''A''''n'' is Abelian Iff ''n'' ≤ 3 and Simple iff ''n'' = 3 or ''n'' ≥ 5. A5 is the smallest non-abelian simple group, having order 60. CONJUGACY CLASSES As in the Symmetric Group , the Conjugacy Classes in A''n'' consist of elements with the same cycle shape. However, if the cycle shape consists of cycles of odd length with no two cycles the same length, then there are exactly two conjugacy classes for this cycle shape. Examples:
AUTOMORPHISM GROUP For ''n'' > 3, except for ''n'' = 6, the Automorphism Group of A''n'' is the symmetric group S''n'', with Inner Automorphism Group A''n'' and Outer Automorphism Group Z2. For ''n'' = 1 and 2, the automorphism group is trivial. For ''n'' = 3 the automorphism group is Z2, with trivial inner automorphism group and outer automorphism group Z2. The outer automorphism group of A6 is Z22. The extra outer automorphism in A6 swaps the 3-cycles (like (123)) with elements of shape 32 (like (123)(456)). EXCEPTIONAL ISOMORPHISMS There are some Isomorphisms between some of the small alternating groups and small Groups Of Lie Type . These are:
More obviously, A3 is isomorphic to the Cyclic Group Z3, and A1 and A2 are isomorphic to the Trivial Group . SUBGROUPS |
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