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It makes more sense to say the fundamental group is isomorphic to Z as a group. It's awkward to say a group is isomorphic to a ring; really, it's isomorphic to the additive group of that ring.
if there exists a [[mapping]] <math>H:[0,1]\times[0,1]\rightarrow X</math> with <math>H(t,0) = \gamma_1(t)</math> and <math>H(t,1) = \gamma_2(t)</math>. We can define an group structure on the resulting [[equivalence class]] of curves by declaring <math>\gamma_1\cdot\gamma_2</math> to be the curve <math>\gamma_2</math> followed by <math>\gamma_1</math>, and rescaled so that the resulting curve still has domain <math>[0,1]</math>. This group is called the fundamental group of <math>X</math>
if there exists a [[mapping]] <math>H:[0,1]\times[0,1]\rightarrow X</math> with <math>H(t,0) = \gamma_1(t)</math> and <math>H(t,1) = \gamma_2(t)</math>. We can define an group structure on the resulting [[equivalence class]] of curves by declaring <math>\gamma_1\cdot\gamma_2</math> to be the curve <math>\gamma_2</math> followed by <math>\gamma_1</math>, and rescaled so that the resulting curve still has domain <math>[0,1]</math>. This group is called the fundamental group of <math>X</math>
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Example: The most important example is the topological space <math>S^1</math>. It can be shown that the fundamental group of <math>S^1</math> is the [[ring_(mathematics)|ring]] of [[integer]]s <math>\mathbf{Z}</math>.
Example: The most important example is the topological space <math>S^1</math>. It can be shown that the fundamental group of <math>S^1</math> is isomorphic to the additive group of [[integer]]s <math>\mathbf{Z}</math>.
[[Category:Topology]]
[[Category:Topology]]
[[Category:Algebra]]
[[Category:Algebra]]