Evariste Galois had a theory of algebraic equations and the key to solving them was symmetry. A man by the name of Marius Sophus Lie wondered if there was a theory for differential equations similar to the theory of algebraic equations. Of course the answer was yes, and once again symmetry was the key. Groups studied by Galois were all finite, however there are many groups that are infinite and include symmetry groups of differential equations.
A common infinite group is the symmetry group of a circle. This group contains transformations that rotate the circle through absolutely any angle. Since the possible angles are infinite, the rotation group of the circle is infinite. This group has the symbol SO(2); the "O" stand for "orthogonal," meaning that the "transformations are rigid motions of the plane"; and the "S" stands for "special," meaning that the "rotations do not flip the plane over." Circles also have an infinite number of axes of reflectional symmetry. A bigge group is formed by adding the reflections, O(2).
Both of these groups are infinite. By specifying the relevant angle, the different rotations can all be determined. When two rotations are composed, you just add the corresponding angles. Lie called this "continuous" and therefore that made the group SO(2) a continuous group. It is also a one dimensional group because only one number is needed to specify the angle. The same applies to the group O(2) because all that is needed is a way to distinguish reflections from rotations which would just consist of changing a plus or minus sign in the algebra.
The left diagram shows that the circle has an infinite numer of rotational symmetries and the right shows that the circle has an infinite number of reflectional symmetries.
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