Editing Dihedral group (section)

===Examples of automorphism groups===
{{math|D{{sub|9}}}} has 18 [[inner automorphism]]s. As 2D isometry group D<sub>9</sub>, the group has mirrors at 20° intervals. The 18 inner automorphisms provide rotation of the mirrors by multiples of 20°, and reflections. As isometry group these are all automorphisms. As abstract group there are in addition to these, 36 [[outer automorphism]]s; e.g., multiplying angles of rotation by 2.

{{math|D{{sub|10}}}} has 10 inner automorphisms. As 2D isometry group D<sub>10</sub>, the group has mirrors at 18° intervals. The 10 inner automorphisms provide rotation of the mirrors by multiples of 36°, and reflections. As isometry group there are 10 more automorphisms; they are conjugates by isometries outside the group, rotating the mirrors 18° with respect to the inner automorphisms. As abstract group there are in addition to these 10 inner and 10 outer automorphisms, 20 more outer automorphisms; e.g., multiplying rotations by 3.

Compare the values 6 and 4 for [[Euler's totient function]], the [[multiplicative group of integers modulo n|multiplicative group of integers modulo ''n'']] for ''n'' = 9 and 10, respectively. This triples and doubles the number of automorphisms compared with the two automorphisms as isometries (keeping the order of the rotations the same or reversing the order).

The only values of ''n'' for which ''φ''(''n'') = 2 are 3, 4, and 6, and consequently, there are only three dihedral groups that are isomorphic to their own automorphism groups, namely {{math|D{{sub|3}}}} (order 6), {{math|D{{sub|4}}}} (order 8), and {{math|D{{sub|6}}}} (order 12).<ref>{{cite book|title=A Course in Group Theory|first1=John F.|last1=Humphreys|year=1996|isbn=9780198534594|publisher=Oxford University Press|page=195|url=https://books.google.com/books?id=2jBqvVb0Q-AC&pg=PA195}}</ref><ref>{{cite web|url=http://www.math.ucsd.edu/~atparris/small_groups.html|title=Groups of small order|first1=John|last1=Pedersen|publisher=Dept of Mathematics, University of South Florida}}</ref><ref>{{cite web|url=http://math.uchicago.edu/~may/REU2013/REUPapers/Sommer-Simpson.pdf |archive-url=https://web.archive.org/web/20160806115458/http://math.uchicago.edu/~may/REU2013/REUPapers/Sommer-Simpson.pdf |archive-date=2016-08-06 |url-status=live|title=Automorphism groups for semidirect products of cyclic groups|first1=Jasha|last1=Sommer-Simpson|page=13|quote='''Corollary 7.3.''' Aut(D<sub>''n''</sub>) = D<sub>''n''</sub> if and only if ''φ''(''n'') = 2|date=2 November 2013}}</ref>