Editing Fundamental group (section)

===Group structure===
[[File:Homotopy_group_addition.svg|Addition of loops|thumb]]
By the above definition, <math>\pi_1(X, x_0)</math> is just a set. It becomes a [[group (mathematics)|group]] (and therefore deserves the name fundamental ''group'') using the concatenation of loops. More precisely, given two loops <math>\gamma_0, \gamma_1</math>, their product is defined as the loop
:<math>\gamma_0 \cdot \gamma_1 \colon [0, 1] \to X</math>
:<math>(\gamma_0 \cdot \gamma_1)(t) = \begin{cases}
        \gamma_0(2t) & 0 \leq t \leq \tfrac{1}{2} \\
    \gamma_1(2t - 1) & \tfrac{1}{2} \leq t \leq 1.
  \end{cases}</math>
Thus the loop <math>\gamma_0 \cdot \gamma_1</math> first follows the loop <math>\gamma_0</math> with "twice the speed" and then follows <math>\gamma_1</math> with "twice the speed".

The product of two homotopy classes of loops <math>[\gamma_0]</math> and <math>[\gamma_1]</math> is then defined as <math>[\gamma_0 \cdot \gamma_1]</math>. It can be shown that this product does not depend on the choice of representatives and therefore gives a [[Equivalence relation#Well-definedness under an equivalence relation|well-defined]] operation on the set <math>\pi_1(X, x_0)</math>. This operation turns <math>\pi_1(X, x_0)</math> into a group. Its [[neutral element]] is the constant loop, which stays at <math>x_0</math> for all times ''t''. The [[inverse element|inverse]] of a (homotopy class of a) loop is the same loop, but traversed in the opposite direction. More formally,
:<math>\gamma^{-1}(t) := \gamma(1-t)</math>.
Given three based loops <math>\gamma_0, \gamma_1, \gamma_2,</math> the product
:<math>(\gamma_0 \cdot \gamma_1) \cdot \gamma_2</math> 
is the concatenation of these loops, traversing <math>\gamma_0</math> and then <math>\gamma_1</math> with quadruple speed, and then <math>\gamma_2</math> with double speed. By comparison,
:<math>\gamma_0 \cdot (\gamma_1 \cdot \gamma_2)</math> 
traverses the same paths (in the same order), but <math>\gamma_0</math> with double speed, and <math>\gamma_1, \gamma_2</math> with quadruple speed. Thus, because of the differing speeds, the two paths are not identical. The [[associativity]] axiom
:<math>[\gamma_0] \cdot \left([\gamma_1] \cdot [\gamma_2]\right) = \left([\gamma_0] \cdot [\gamma_1]\right) \cdot [\gamma_2]</math>
therefore crucially depends on the fact that paths are considered up to homotopy. Indeed, both above composites are homotopic, for example, to the loop that traverses all three loops <math>\gamma_0, \gamma_1, \gamma_2</math> with triple speed. The set of based loops up to homotopy, equipped with the above operation therefore does turn <math>\pi_1(X, x_0)</math> into a group.