# Transposition in symmetric group:S3

From Groupprops

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 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:

## Arithmetic functions

Function | Value | Explanation |
---|---|---|

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 |