Question

2. Define a function R R2 + R2 by 0 · x = Rox, where R_(cos – sin o) R0 = 0 (sin cos 0 ) (a) Prove that this defines a left group action. (b) Let x € R2 – {(0,0)}. Describe geometrically Oz. (c) Compute Gr.

Answer 1

Left group Action Let 'a be a group, and 's' be a non-empty set. A left action a on s is a function 'GxSys, usually denoted (8,5) = gs such that (4) (8182) 5 = 8,(825) 6) es= 5; where e is the indentity of 6 for all ses, 8,82 EA, con In this problem. R plays the role of G C R +) is and IR? is treated as S. L groupe I Now A mapping 8.x = ( caso sino V ). and 2. (89) - (a) This defines a left group action:- (9) ret 80ER I cos (0+6) -sin (0+8)) 10+8)(x) = | sin (0+8) cos(0+0) 30 O., cose – sind, sino cosd) coso , sino Il cosd , sind = | sino cose | sind cosd. I cosa cos & - sino sind . sin /0+8). sin co to) - sino sing this dis e cos (6+6) - sin (8+6) le sin/6+6) cos(a+d))

- (0+6)x= 0 (6x) so get condition is salisfied. (2) Now element of So 0.x= DER is the identity R. mldorg ring coso sino , ( sino caso, To, x = x Hence the mapping satisfies both the conditions. so it defines a left group action. (b) Let de R²-{(0,0)} .Description of Ox oa, denotes the orbit containing a. Da={bes 6=ga for some ge ? Here G=R, and S=R? so on = f berr? b=ga for some ger?

| Let b = (B) and a= (2) so boa PER. COS O -sino ces & 02 cosa a, sind az, sinoa, + cose az Diamo Now, B9 + b3 = (cose a-sino ang? +(sinoa,+ cosa a)? - the distance of b from origin is same as that of an so, la contains all those point which has same distance from origin as a has. So, Oa is a circle whose sodius is of a from the distance the origin

IR 9 (9,,a) (c) compute Ax Gee {gEG 84-x} Gax is called the stabilizer of e. Now ge=x > Ox=r ; g = 0 EIR I cose sine cos sino :) - G) [v=Q)er] sino Coso cose-1 -sine sino coso-)

csino COSO - sino Tzero matrix - toso-1) I this bold; because ato VREIR? the above relation holds]. » Taking determinant, (100)] (case-1) + sin?o=0 y - Coze = 0 » sine 50 3 Bano net] 310=2015) so, Gxt {2010 [nex}