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Cyclic groups Zp Simplicity: Simple for p a prime number. Order: p Schur multiplier: Trivial. Outer automorphism group: Cyclic of order p − 1. Other names: Z/pZ Remarks: These are the only simple groups that are not perfect. An, n > 4, Alternating groups Simplicity: Solvable for n < 5, otherwise simple. Order: n!/2 when n > 1. Schur multiplier: 2 for n = 5 or n > 7, 6 for n = 6 or 7; see Covering groups of the alternating and symmetric groups Outer automorphism group: In general 2. Exceptions: for n = 1, n = 2, it is trivial, and for n = 6, it has order 4 (elementary abelian). Other names: Altn. There is an unfortunate conflict with the notation for the (unrelated) groups An(q), and some authors use various different fonts for An to distinguish them. In particular, in this article we make the distinction by setting the alternating groups An in Roman font and the Lie-type groups An(q) in italic. Isomorphisms: A1 and A2 are trivial. A3 is cyclic of order 3. A4 is isomorphic to A1(3) (solvable). A5 is isomorphic to A1(4) and to A1(5). A6 is isomorphic to A1(9) and to the derived group B2(2)'. A8 is isomorphic to A3(2). Remarks: An index 2 subgroup of the symmetric group of permutations of n points when n > 1. |
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