Editing Group isomorphism (section)

== Cyclic groups ==
All cyclic groups of a given order are isomorphic to <math>(\Z_n, +_n),</math> where <math>+_n</math> denotes addition [[modular arithmetic|modulo]] <math>n.</math>

Let <math>G</math> be a cyclic group and <math>n</math> be the order of <math>G.</math> Letting <math>x</math> be a generator of <math>G</math>, <math>G</math> is then equal to <math>\langle x \rangle = \left\{e, x, \ldots, x^{n-1}\right\}.</math> 
We will show that
<math display="block">G \cong (\Z_n, +_n).</math>

Define 
<math display="block">\varphi : G \to \Z_n = \{0, 1, \ldots, n - 1\},</math> so that <math>\varphi(x^a) = a.</math> 
Clearly, <math>\varphi</math> is bijective. Then
<math display="block">\varphi(x^a \cdot x^b) = \varphi(x^{a+b}) = a + b = \varphi(x^a) +_n \varphi(x^b),</math> 
which proves that <math>G \cong (\Z_n, +_n).</math>