Classification of rational dihedral groups
From Groupprops
Statement
The following are equivalent for a natural number :
- The Dihedral group (?) of order
(and degree
) is a Rational group (?): all its characters over the complex numbers are rational-valued.
- The dihedral group of order
and degree
is a Rational-representation group (?): all its representations over the complex numbers can be realized over the rational numbers.
-
, where
denotes the Euler phi-function, i.e., the order of the multiplicative group of integers modulo
.
-
.
Thus, the five dihedral groups that are rational are:
-
: cyclic group:Z2 (a degenerate case)
-
: Klein four-group (also a degenerate case)
-
: Isomorphic to symmetric group:S3.
-
: dihedral group:D8
-
:
, isomorphic to direct product of S3 and Z2