# Isomorphism class

• In homotopy theory, the fundamental group of a space ${\displaystyle X}$ at a point ${\displaystyle p}$, though technically denoted ${\displaystyle \pi _{1}(X,p)}$ to emphasize the dependence on the base point, is often written lazily as simply ${\displaystyle \pi _{1}(X)}$ if ${\displaystyle X}$ is path connected. The reason for this is that the existence of a path between two points allows one to identify loops at one with loops at the other; however, unless ${\displaystyle \pi _{1}(X,p)}$ is abelian this isomorphism is non-unique. Furthermore, the classification of covering spaces makes strict reference to particular subgroups of ${\displaystyle \pi _{1}(X,p)}$, specifically distinguishing between isomorphic but conjugate subgroups, and therefore amalgamating the elements of an isomorphism class into a single featureless object seriously decreases the level of detail provided by the theory.