Template:Particular automorphism class
We consider the group
defined as symmetric group:S3, i.e., the symmetric group of degree three, which for convenience we take to be the symmetric group acting on the set
.
We are interested in the conjugacy class of 3-cycles in this group, i.e., permutations that move all elements and have order three. The 3-cycles form a single conjugacy class and also form a single orbit under the action of the automorphism group of
.
The full list of elements in the conjugacy class is:
Arithmetic functions
Function |
Value |
Explanation
|
order of the whole group |
6 |
|
size of conjugacy class |
2 |
|
size of automorphism class |
2 |
|
number of conjugacy classes in automorphism class |
1 |
|
order of elements in conjugacy class |
3 |
|
Description in alternative interpretations of the whole group
Interpretation of |
Description of conjugacy class |
Verification of size computation |
Verification of element order computation |
More information
|
As the symmetric group of degree three |
corresponds to the partition 3 |
|
|
element structure of symmetric groups, element structure of symmetric group:S3#Interpretation as symmetric group, conjugacy class size formula in symmetric group
|
As the dihedral group , ( odd) |
the generator of the cyclic piece of order |
2 (valid for ) |
|
element structure of dihedral groups, element structure of symmetric group:S3#Interpretation as dihedral group
|
As the special linear group , ( even) |
the unique conjugacy class of semisimple elements not diagonalizable over the field itself |
|
Must divide in general, in this case, it is |
element structure of special linear group of degree two over a finite field, element structure of symmetric group:S3#Interpretation as general linear group of degree two
|
As the general affine group , |
the conjugacy class of all maps of the form . |
|
|
element structure of general affine group of degree one over a finite field, element structure of symmetric group:S3#Interpretation as general affine group of degree one
|