2085Q4 Show that the following equation defines a group acti

208_5Q4

Show that the following equation defines a group action of (R^*,x) on the plane R^2: c ^ (x, y) = (x, cy). The remainder of this question concerns the group action given in part (a). Determine the orbits of (1,0) (0,1) (-1,1) Give a geometric description of ALL the orbits of the action. Determine the stabilisers of (1,0), (0,1), Determine the fixed set Fix (3).

Solution

a) We show two things:
(i) 1 /\\ (x,y) = (x, y) for all x,y in R.
This is easy to see: 1 /\\ (x,y) = (x, 1*y) = (x, y).

(ii) (cs)/\\ (x,y) = c /\\ [s /\\ (x,y)] for all c,s in R^+ and x,y in R.
(cs)/\\ (x,y) = (x, (cs)y)
= (x, c(sy))
= c /\\ (x, sy)
= c /\\ [s /\\ (x,y)].

Thus, we have a group action.

bc) orbit(x,y) = { c /\\ (x,y) = (x,cy) | c in R^+}.

So, orbit(1,0) = {(1,0)}, just a point.
orbit(0,1) = {(0,c) | c in R^+}, the y-axis
orbit(1,-1) = {(1,-c) | c in R^+}., the line x = 1.

d) stab(1,0) = {c in R^+ | c /\\ (1,0) = (1,0)}
= {c in R^+ | (1, c * 0) = (1,0)}
= {c in R^+}

stab(0,1) = {c in R^+ | c /\\ (0,1) = (0,1)}
= {c in R^+ | (0,c) = (0,1)}
= {1}.


e) Fix(3) = {(x,y) | 3 ^(x,y) = (x,y) for all x,y in R}
= {(x,y) | (x,3y) = (x,y) for all x,y in R}.
Thus, we need y = 0. (x can be anything)
= {(x,0)| x in R}.