¹Department of Mathematics, CU Shah University, Wadhwan City-363 030 Gujarat (India)
²Department of Mathematics, Gujarat University Ahmedabad-380 009 Gujarat (India)

In this article we study symmetries of plane figures. We start with the definition of isometry of a metric space. For the better understanding of all the isometries of R2, we present the known results in form of a flow chart. And then using this idea of isometry, we define symmetry group of a subset of R2. With the use of this definition we determine symmetry groups of the capital Alphabets of English. We also describe the symmetry group of a regular n-gon which is known as
Dn. We also describe subsets of R2 where symmetry group is the cyclic group of order n which is denoted usually by Cn