In an earlier post I decomposed the permutation module for acting on the non-zero lines in , with coefficients in a field of characteristic . Some related ideas give the shortest proof I know of the exceptional isomorphism between and . The motivation section below explains why what we do ‘must work’, but is not logically necessary.

### Motivation

Let have order and let

The simple modules for in characteristic are the trivial module, the natural module , its dual , and the reduction modulo of a module affording the -dimensional character induced from either of the two non-trivial linear characters of . (A short calculation with Mackey’s Formula and Frobenius Reciprocity shows the module is irreducible in characteristic zero; then since is odd, it reduces modulo to a simple projective. Alternatively one can work over and perform the analogue of the characteristic zero construction directly.) The permutation module for acting on the cosets of , defined over , is projective. Since it has the trivial module in its socle and top, the only possible Loewy structures are

In every case, the permutation module contains a -dimensional -submodule having either or as a top composition factor. Below we construct as a module for and hence obtain the exceptional isomorphism in the form .

In fact the final two Loewy structures can be ruled out because they are asymmetric with respect to and , and so paired under the outer automorphism of . Thus if one exists then so does the other, and there would be two different projective covers of the trivial module (impossible).

### Construction

Let regarded as a permutation group acting on the non-zero lines in . For , let denote the line through , and let denote the line through . Let be the corresponding -permutation module.

Let , corresponding to the Möbius map and to the matrix in . The simple modules for correspond to the factors of

Since is permuted by squaring, an idempotent killing the simple modules with minimal polynomial is . Therefore the module we require has codimension in the -dimensional module

Here is the sum of the first three basis vectors, so in its right action, is represented by

.

By symmetry must appear in some vector in the -module we require. It therefore seems a plausible guess that we require is generated by where

and so has basis where for each .

Let and let

chosen so that and . The corresponding Möbius maps are and and one choice of corresponding matrices in is and . Extending the calculation for above shows that in their (right) action on ,

The unique trivial submodule of is spanned by

Quotienting out, we obtain the -dimensional representation of where

Since has cyclic Sylow -subgroups and cyclic Sylow -subgroups, and a unique conjugacy class of elements of order (which contains ), one can see just from the matrices above that the representation is faithful. Comparing orders, we get .