# Element structure of projective special linear group:PSL(3,2)

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This article gives specific information, namely, element structure, about a particular group, namely: projective special linear group:PSL(3,2).

View element structure of particular groups | View other specific information about projective special linear group:PSL(3,2)

This article describes the element structure of projective special linear group:PSL(3,2), which is the same as , . It is also isomorphic to the projective special linear group of degree two over field:F7, i.e., the group .

## Summary

Item | Value |
---|---|

order of the whole group (total number of elements) | 168 |

conjugacy class sizes | 1,21,24,24,42,56 in grouped form: 1 (1 time), 21 (1 time), 24 (2 times), 42 (1 time), 56 (1 time) maximum: 56, number of conjugacy classes: 6, lcm: 168 |

order statistics | 1 of order 1, 21 of order 2, 56 of order 3, 42 of order 4, 48 of order 7 maximum: 7, lcm (exponent of the whole group): 84 |

## Conjugacy class structure

### Interpretation as projective special linear group of degree two

Compare with element structure of projective special linear group of degree two over a finite field#Conjugacy class structure

We consider the group as , . We use the letter to denote the generic case of .

Nature of conjugacy class upstairs in | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class (generic that is 3 mod 4) | Size of conjugacy class () | Number of such conjugacy classes (generic that is 3 mod 4) | Number of such conjugacy classes () | Total number of elements (generic that is 3 mod 4) | Total number of elements () |
---|---|---|---|---|---|---|---|---|---|

Diagonalizable over with equal diagonal entries, hence a scalar | or , both correspond to the same element | where | where | 1 | 1 | 1 | 1 | 1 | 1 |

Diagonalizable over , not over , eigenvalues square roots of | Square roots of | 21 | 1 | 1 | 21 | ||||

Not diagonal, has Jordan block of size two | (multiplicity 2) or (multiplicity 2). Each conjugacy class has one representative of each type. | where | where | 24 | 2 | 2 | 48 | ||

Diagonalizable over , not over . Must necessarily have no repeated eigenvalues. Eigenvalues not square roots of . | Pair of conjugate elements of of norm 1, not square roots of -1. Each pair identified with its negative pair. | , irreducible; note that 's pair and 's pair get identified. | Same as characteristic polynomial | 42 | 1 | 42 | |||

Diagonalizable over with distinct (and hence mutually inverse) diagonal entries |
where where are square roots of . Note that the representative pairs and get identified. | , again with identification. | , again with identification. | 56 | 1 | 56 | |||

Total | NA | NA | NA | NA | NA | 6 | 168 |

### Interpretation as general linear group of degree three over field:F2

Compare with element structure of general linear group of degree three#Conjugacy class structure. Since many cases (namely, those that rely on distinct eigenvalues) do not arise over field:F2, the cases have been omitted in the table below.

Nature of conjugacy class | Eigenvalues | Characteristic polynomial | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Semisimple? | Diagonalizable over ? |
---|---|---|---|---|---|---|---|---|

Diagonalizable over with equal diagonal entries, hence a scalar | 1 | 1 | 1 | Yes | Yes | |||

Diagonalizable over , not over | Distinct Galois conjugate triple of elements in . If one of the elements is , the other two are and . | irreducible degree three polynomial over | same as characteristic polynomial | 24 | 2 | 48 | Yes | No |

One eigenvalue is in , the other two are in | one element of , pair of Galois conjugates over in . | product of linear polynomial and irreducible degree two polynomial over | same as characteristic polynomial | 56 | 1 | 56 | Yes | No |

Has Jordan blocks of sizes 2 and 1 with equal eigenvalues over | with | 42 | 1 | 42 | No | No | ||

Has Jordan block of size 3 | with | same as characteristic polynomial | 21 | 1 | 21 | No | No | |

Total | NA | NA | NA | NA | 6 | 168 | NA | NA |