The automorphism group of the doubly-even [72,36,16] code can only be of order 1, 3 or 5

We prove that a putative $[72,36,16]$ code is not the image of linear code over $\ZZ_4$, $\FF_2 + u \FF_2$ or $\FF_2+v\FF_2$, thus proving that the extremal doubly even $[72,36,16]$-binary code cannot have an automorphism group containing a fixed point-free involution. Combining this with the previously proved result by Bouyuklieva that such a code cannot have an automorphism group containing an involution with fixed points, we conclude that the automorphism group of the $[72,36,16]$-code cannot be of even order, leaving 3 and 5 as the only possibilities.