Transposition in symmetric group:S3
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 transpositions in this group, i.e., permutations that flip two elements and fix the third element. The transpositions 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:
|order of the whole group||6|
|size of conjugacy class||3|
|size of automorphism class||3|
|number of conjugacy classes in automorphism class||1|
|order of elements in conjugacy class||2|
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 2 + 1||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 conjugacy class of all "reflections" i.e. things outside the cyclic subgroup of index two||2||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 unipotent elements (note that the conjugacy class is unique when is even)||characteristic prime, i.e., prime whose power is . In this case, it is 2||element structure of special linear group of degree two over a finite field, element structure of symmetric group:S3#Interpretation as special linear group of degree two|
|As the general affine group ,||the conjugacy class of all maps of the form .||order of 2 in , which equals 2||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|
|As the general semilinear group ,||the conjugacy class of all maps whose field automorphism part is the non-identity map. Note that for generic , there are such conjugacy classes, but since , the conjugacy class is unique.||Since this is the conjugacy class containing the mapping that is simply the field automorphism, it has order 2.||element structure of general semilinear group of degree one over a finite field, element structure of symmetric group:S3#Interpretation as general semilinear group of degree one|