#### Groups

The equivalence relation induced by homotopy starts to enter the realm of algebraic topology, which is a branch of mathematics that characterizes the structure of topological spaces in terms of algebraic objects, such as groups. These resulting groups have important implications for motion planning. Therefore, we give a brief overview. First, the notion of a group must be precisely defined. A group is a set, , together with a binary operation, , such that the following group axioms are satisfied:

1. (Closure) For any , the product .
2. (Associativity) For all , . Hence, parentheses are not needed, and the product may be written as .
3. (Identity) There is an element , called the identity, such that for all , and .
4. (Inverse) For every element , there is an element , called the inverse of , for which and .
Here are some examples.

Example 4..7 (Simple Examples of Groups)   The set of integers is a group with respect to addition. The identity is 0, and the inverse of each is . The set of rational numbers with 0 removed is a group with respect to multiplication. The identity is , and the inverse of every element, , is (0 was removed to avoid division by zero).

An important property, which only some groups possess, is commutativity: for any . The group in this case is called commutative or Abelian. We will encounter examples of both kinds of groups, both commutative and noncommutative. An example of a commutative group is vector addition over . The set of all 3D rotations is an example of a noncommutative group.

Steven M LaValle 2012-04-20