Galois groups, in particular simple Galois groups, play a fundamental role in the TGD view of cognition. The TGD based model of the genetic code involves in an essential manner the groups A5 (icosahedron), which is the smallest non-abelian simple group, and A4 (tetrahedron). The identification of these groups as Galois groups leads to a more precise view about genetic code. The question why the genetic code is a fusion of 3 icosahedral codes and of only a single tetrahedral code remained however poorly understood.
The identification of the symmetry groups of the I, O, and T as Galois groups makes it possible to answer this question. Icosa-tetrahedral tesselation of 3-D hyperbolic space H3, playing centrl role in TGD, can be replaced with its 3-fold covering replacing I/O/T with the corresponding symmetry group acting as a Galois group. T has only only a single Hamiltonian cycle and its 3-fold covering behaves effectively as a single cycle. Octahedral codons can be regarded as icosahedral and tetrahedral codons so they do not contribute to the code.
See the article Galois groups and genetic code.
For a summary of earlier postings see Latest progress in TGD.