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
Famous quotes containing the word exceptional:
“The universal social pressure upon women to be all alike, and do all the same things, and to be content with identical restrictions, has resulted not only in terrible suffering in the lives of exceptional women, but also in the loss of unmeasured feminine values in special gifts. The Drama of the Woman of Genius has too often been a tragedy of misshapen and perverted power.”
—Anna Garlin Spencer (1851–1931)