Editing Group isomorphism (section)

== Examples ==
In this section some notable examples of isomorphic groups are listed.
* The group of all [[real number]]s under addition, <math>(\R, +)</math>, is isomorphic to the group of [[positive real numbers]] under multiplication <math>(\R^+, \times)</math>:
*:<math>(\R, +) \cong (\R^+, \times)</math> via the isomorphism <math>f(x) = e^x</math>.
* The group <math>\Z</math> of [[integer]]s (with addition) is a subgroup of <math>\R,</math> and the [[factor group]] <math>\R/\Z</math> is isomorphic to the group <math>S^1</math> of [[complex number]]s of [[absolute value]] 1 (under multiplication):
*:<math>\R/\Z \cong S^1</math>
* The [[Klein four-group]] is isomorphic to the [[Direct product of groups|direct product]] of two copies of <math>\Z_2 = \Z/2\Z</math>, and can therefore be written <math>\Z_2 \times \Z_2.</math> Another notation is <math>\operatorname{Dih}_2,</math> because it is a [[dihedral group]].
* Generalizing this, for all [[parity (mathematics)|odd]] <math>n,</math> <math>\operatorname{Dih}_{2 n}</math> is isomorphic to the direct product of <math>\operatorname{Dih}_n</math> and <math>\Z_2.</math>
* If <math>(G, *)</math> is an [[infinite cyclic group]], then <math>(G, *)</math> is isomorphic to the integers (with the addition operation). From an algebraic point of view, this means that the set of all integers (with the addition operation) is the "only" infinite cyclic group.

Some groups can be proven to be isomorphic, relying on the [[axiom of choice]], but the proof does not indicate how to construct a concrete isomorphism. Examples:
* The group <math>(\R, +)</math> is isomorphic to the group <math>(\Complex, +)</math> of all complex numbers under addition.<ref>{{cite journal|last1= Ash|year=1973|title=A Consequence of the Axiom of Choice|journal=Journal of the Australian Mathematical Society|volume=19|issue=3|pages=306–308|doi=10.1017/S1446788700031505|url=http://journals.cambridge.org/download.php?file=%2FJAZ%2FJAZ1_19_03%2FS1446788700031505a.pdf&code=d2e5b0d7bbbbe7368eb4aa14d4bda045|access-date=21 September 2013|doi-access=free}}</ref>
* The group <math>(\Complex^*, \cdot)</math> of non-zero complex numbers with multiplication as the operation is isomorphic to the group <math>S^1</math> mentioned above.