Question

Hello,

I would like to solve 4.5 step by step please

4.5 Let G be finite. Show that the probability that two elements of G chosen at random commute with each other is equal to k/G], where k is the number of conjugacy classes of G. (The two elements are allowed to be equal; they are chosen randomly with replacement.) 4.6 Find a formula for the number of fundamentally different ways to color the six fel. Tidad

Answer 1

Solution Le P(G) depote the probability of . the every that selected touro elemery commute each other. Where G is finite group of order Gl. Total no. of choices of selecting a elemeny randomly es = GIGI = 1G12 Non consider a set o k={ (XY) EGOG 1 xy=you} Then P (G) = IKI Now for every x EG (49) Ek iff y + C (x) ( 33-3) LIK) = [ 1 CCL)) Now Chooping one representative from every Conjugery clus, let say ry, H2,. . . . . . Then IKI = Iccost = Ž G: Cl] I Clail TG 12 LEG sony [G: CRCC] = 16 Como = 1611 - kIGI KIG) - k where he is number PG 1612 7 corgugely class en of