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, $G$, together with a binary operation, $\circ$, such that the following group axioms are satisfied:

1. (Closure) For any $a, b \in G$, the product $a \circ b \in G$.
2. (Associativity) For all $a, b, c \in G$, $(a \circ b) \circ c = a \circ (b \circ c)$. Hence, parentheses are not needed, and the product may be written as $a \circ b \circ c$.
3. (Identity) There is an element $e \in G$, called the identity, such that for all $a \in G$, $e \circ a = a$ and $a \circ e = a$.
4. (Inverse) For every element $a \in G$, there is an element $a^{-1}$, called the inverse of $a$, for which $a \circ a^{-1} = e$ and $a^{-1} \circ a = e$.

Here are some examples.

Example 4..7 (Simple Examples of Groups)   The set of integers ${\mathbb{Z}}$ is a group with respect to addition. The identity is 0, and the inverse of each $i$ is $-i$. The set ${\mathbb{Q}} \setminus 0$ of rational numbers with 0 removed is a group with respect to multiplication. The identity is $1$, and the inverse of every element, $q$, is $1/q$ (0 was removed to avoid division by zero). $\blacksquare$

An important property, which only some groups possess, is commutativity: $a \circ b = b \circ a$ for any $a, b \in G$. 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 ${\mathbb{R}}^n$. The set of all 3D rotations is an example of a noncommutative group.