Alternating Group - Exceptional Isomorphisms

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).