The binary icosahedral group previous next

The 60-element group A5 ⊂ SO(3) is double covered by
an 120-element group Γ ⊂ SU(2). What is this group?

It's not S5. It's also not A5 × ℤ/2.

It's called the binary icosahedral group.

Just as Klein looked for rational functions on ℂP1 invariant under A5,
we can look for polynomials on ℂ2 invariant under Γ.











To learn more, read the Wikipedia articles on the binary icosahedral group and icosahedral symmetry: commonly confused groups. In fact we have

Γ ≅ SL(2,𝔽5)

that is, 2 × 2 matrices with determinant 1 having entries in the field with 5 elements, while

A5 ≅ PSL(2,𝔽5)

that is, SL(2,𝔽5) modulo ±I.