Let K be the subgroup r0 v of D4 Show that r1 t mod K but r2

Let K be the subgroup {r_0, v} of D_4. Show that r_1 t (mod K), but r_2 h (mod K), but r_1 r_2 t h (mod K).

Solution

It is assumed D₄ is the dihedral group --symmetries of the square .(The notation used in the problem is not clear)

D₄ = {I, S,S², S³ ,R, RS,RS², RS³ }, where S is a rotation by 90* and R is a reflection satisfying the relation

                                                    RSR = S³ =S^-1.

It is required to prove that the subgroup K ={I, R} is not normal in D₄. (Again , it is not clear what the symbols t, h, r_i stand for in the problem)

Suffices to show that S does not normalize this subgroup K.

Consider SRS^-1 =( RS^-1 ) S^-1 =S^-2 , which does not belong to K.

So K is not normal in D₄