Let K be the subgroup r0 v of D4 Show that r1 equivalence t

Let K be the subgroup {r_0, v} of D_4. Show that r_1 equivalence t (mod K0 and t_2 equivalence h (mod K0, but r_1 o r_2 not equal sign t o h 9mod K).

Solution

The notation has not been explained in the problem.

We use the following notation.

Let D₄ be the dihedral group

                       D₄ = {e,R,R² ,R³ , T,TR,TR² ,TR³ }

generated by the rotation R and reflection T satisfying the following relations:

             R⁴ = T² =e and TRT= R^-1 =R³ ....................................................(1)

K = {e,T}

Problem statement is to show that the subgroup K is not normal in D_4.

Suffices to show that R does not normalize K.

Consider the conjugate RTR^-1 = TR³ R^-1 = TR² , which is neither identity nor T.

Hence the result.