Question

Let D4 be the group of symmetries of the square That is, D4 = {1, R, R2, Rº, T., Ty, T1,3, T2,4} where, in particular, R is a counterclockwise rotation by 90° about the origin and Tx is a reflection about the x-axis (the group and its elements were defined in class). (a) Show that D4 is generated by {R, Tx}, that is, D4 = (R, Tx). (b) Construct the Cayley graph Cay(D4, {R, Tx}).

Answer 1

Square / Solution (o: let D4 be a group of symmetry of the squ let Du = {I, R, R3 R3 TH. Ty, T1, 3, 724}; tet we show that Da is generated by { R, Tx} ie Du = {R Tx let s= {R, Tx} Then <s>=D4 ist every element of Decembe written as finite product of elements of S and their inverses. So let us express, (s) explicitly by, (3>={TR" I minezzi since R is rotation by goo in counterclockwise direction, so as, R, Tx E DA So By closure prop., Tumorn € Du & minth so (s>C DA Now, let RR E D4 ( OSK (3) Then I= TXORO R = TROR R² = TXOR?, R²²Tx o R3 and similarly, The THOR Ty = TXOR, T, 3 = Too R², To A = Txo R3, so, as every -
to
den element JD4 can be generated by elements of Ri Tx and so, & 26D4) at <s> -) Du CLS] to Now, from Equ ( klii, we conclude that Du= <s> and so we can conclude that {Du = Ls) ={R. Tx> Rond. 2 can

(6) cayley graph for (Du , {R. Tm3) can be drawenas The follows: WTOR3 THORA of Txor Y - R3 The Cayley diagraph of D4., Tyk R3 The Cayley graph of D4