Exceptional Isomorphisms
There are some exceptional isomorphisms between some of the small alternating groups and small groups of Lie type, particularly projective special linear groups. These are:
- A4 is isomorphic to PSL2(3) and the symmetry group of chiral tetrahedral symmetry.
- A5 is isomorphic to PSL2(4), PSL2(5), and the symmetry group of chiral icosahedral symmetry.(See for an indirect isomorphism of using a classification of simple groups of order 60, and here for a direct proof).
- A6 is isomorphic to PSL2(9) and PSp4(2)'
- A8 is isomorphic to PSL4(2)
More obviously, A3 is isomorphic to the cyclic group Z3, and A0, A1, and A2 are isomorphic to the trivial group (which is also SL1(q)=PSL1(q) for any q).
Read more about this topic: Alternating Group
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