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Element structure of alternating group:A4 - Groupprops

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The alternating group of degree four has order 12, with prime factorization . Below are listed various methods that can be used to compute the order, all of which should give the answer 12: For a symmetric group, cycle type determines conjugacy class. The statement is almost true for the alternating group, except for the fact that some conjugacy classes of even permutations in the symmetric group split into two in the alternating group, as per the splitting criterion for conjugacy classes in the alternating group, which says that a conjugacy class of even permutations splits in the alternating group if and only if its cycle decomposition comprises odd cycles of distinct length. Here are the unsplit conjugacy classes: In this case, the union of the unsplit conjugacy classes is a proper normal subgroup isomorphic to the Klein four-group. Note that this phenomenon is unique to the case . Here is the split conjugacy class: We consider the group as , . We use the letter to denote the gener

Element structure of alternating group:A4 - Groupprops Jump to content Want site search autocompletion? See here Encountering 429 Too Many Requests errors when browsing the site? See here From Groupprops This article gives specific information, namely, element structure , about a particular group , namely: alternating group:A4 . View element structure of particular groups | View other specific information about alternating group:A4 This article gives information on the element structure of alternating group:A4 . See also element structure of alternating groups and element structure of symmetri

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