Small Dihedral Groups
For n = 1 we have Dih1. This notation is rarely used except in the framework of the series, because it is equal to Z2. For n = 2 we have Dih2, the Klein four-group. Both are exceptional within the series:
- They are abelian; for all other values of n the group Dihn is not abelian.
- They are not subgroups of the symmetric group Sn, corresponding to the fact that 2n > n ! for these n.
The cycle graphs of dihedral groups consist of an n-element cycle and n 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups stand for the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the identity element.
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