Answer to Question #47325 in Abstract Algebra for ayush kumar

Question

2a) An isometry of R2 is a map 2 2 :R R which preserves the Euclidean distance between any
two points in R2, i.e., (x)  (y)  x  y x, y R2.
It is known that if (0,0)  (0,0), then  is a linear invertible map, and is called a linear
isometry.
Prove that the set L of linear isometries of R2 is a group with respect to the composition of
functions.
b) Let (G, ) be a group with 5 elements. Construct a Cayley table for . How many distinct
Cayley tables can you construct ?