Examples
The dihedral group of order 10 has two Nielsen equivalence classes of generating sets of size 2. Letting x be an element of order 2, and y being an element of order 5, the two classes of generating sets are represented by and, and each class has 15 distinct elements. A very important generating set of a dihedral group is the generating set from its presentation as a Coxeter group. Such a generating set for a dihedral group of order 10 consists of any pair of elements of order 2, such as . This generating set is equivalent to via the complicated:
- , type 3
- , type 1
- , type 3
- , type 4
- , type 3
- , type 1
- , type 3
Unlike and, the generating sets and are equivalent. A transforming sequence using more convenient elementary transformations (all swaps, all inverses, all products) is:
- , multiply 2nd generator into 3rd
- , multiply 3rd generator into 2nd
- , multiply 2nd generator into 3rd
- , multiply 2nd generator into 3rd
Read more about this topic: Nielsen Transformation
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