10
$\begingroup$

I need examples of problems which use, directly or indirectly, the homomorphism $S_4\to S_3$ in the solution (its kernel is $\mathbb{Z}_2\oplus\mathbb{Z}_2$).

Obvious candidates:

  • Lagrange resolvent (the reduction of quartic to cubic equations).

  • Tait's theorem on equivalence of 4-coloring of normal map and 3-coloring of its edges.

Do you have more interesting examples?

$\endgroup$
  • 2
    $\begingroup$ constructing character table of $S_4$? $\endgroup$ – user91132 Jan 17 '16 at 14:50
13
$\begingroup$

I use this or something equivalent in teaching projective geometry to show that the cross-ratio has at most 6 distinct values (when the points are permuted) as opposed to the 24 naively expected. This involves checking that the elements in the Klein 4-group act trivially on the cross-ratio $R(A,B,C,D)$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.