Describe the symmetry group of a line segment viewed as a su

Describe the symmetry group of a line segment viewed as a subset of

a) R^2

b) R^3

Hint: the symmetry group of a line segment considered as a set of points in R2 has order 4, and the symmetry group of a line segment viewed as a set of points in R3 has infinite order

Solution

a) let (x,0) where x lies between -1 to 1 be a line segment of R^2.

symmetry group of the above line segment will be D2.

so it is isomorphic to Z2 xor Z2

b) again proceeding as above let let (x,0,0) where x lies between -1 to 1 be a line segment of R^3.

let G be the symmtric group

it follows that G has fixed the set (-1,0,0 ), (1,0,0)

let S belong to the above set then any element will preserve the relation