# Unitriangular matrix group:UT(4,p)

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.

View other such prime-parametrized groups

## Contents

## Definition

The cases (see unitriangular matrix group:UT(4,2)) and (see unitriangular matrix group:UT(4,3)) differ somewhat from the cases of other primes. This is noted at all places in the page where it becomes significant.

### As a group of matrices

Given a prime , the group is defined as the unitriangular matrix group of degree four over the prime field . Explicitly, this is described as the following group under matrix multiplication:

The multiplication of matrices and gives the matrix where:

The identity element is the identity matrix, i.e., the matrix where all off-diagonal entries are zero and all diagonal entries are 1.

The inverse of a matrix is the matrix where:

### In coordinate form

We may define the group as the set of ordered 6-tuples over (the coordinates are allowed to repeat), with the multiplication law, identity element, and inverse operation given by:

The matrix corresponding to the 6-tuple is:

This definition clearly matches the earlier definition, based on the rules of matrix multiplication.

## Families

These groups fall in the more general family of unitriangular matrix groups. The unitriangular matrix group can be described as the group of unipotent upper-triangular matrices in , which is also a -Sylow subgroup of the general linear group . This further can be generalized to where is the power of a prime . is the -Sylow subgroup of .

## Element structure

`Further information: element structure of unitriangular matrix group:UT(4,p)`

### Summary

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

number of conjugacy classes | equals number of irreducible representations. See number of irreducible representations equals number of conjugacy classes, linear representation theory of unitriangular matrix group of degree four over a finite field |

order | Follows from the general formula, order of is |

conjugacy class size statistics | 1 ( times), ( times), ( times), ( times) |

order statistics | Case : order 1 (1 element), order 2 (27 elements), order 4 (36 elements) Case : order 1 (1 element), order 3 (512 elements), order 9 (216 elements) Case : order 1 (1 element), order ( elements) |

exponent | if if |

### Conjugacy class structure

For the right-most column for the type of matrix, we use the (row number, column number) notation for matrix entries. Explicitly, the matrix under consideration is:

The subgroups mentioned in the table below are:

Subgroup | Visual description | Condition | Order |
---|---|---|---|

center | |||

derived subgroup | |||

unique abelian subgroup of maximum order |

Nature of conjugacy class | Jordan block size decomposition | Minimal polynomial | Size of conjugacy class | Number of such conjugacy classes | Total number of elements | Order of elements in each such conjugacy class | Type of matrix (constraints on ) |
---|---|---|---|---|---|---|---|

identity element | 1 + 1 + 1 + 1 | 1 | 1 | 1 | 1 | all the are zero | |

non-identity element, but central (has Jordan blocks of size 1,1,2 respectively) | 2 + 1 + 1 | 1 | , all the others are zero | ||||

non-central but in derived subgroup, has Jordan blocks of size 1,1,2 | 2 + 1 + 1 | Among and , exactly one of them is nonzero. may be zero or nonzero | |||||

non-central but in derived subgroup, Jordan blocks of size 2,2 | 2 + 2 | Both and are nonzero. may be zero or nonzero | |||||

outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 1,1,2 | 2 + 1 + 1 | is nonzero and are arbitrary | |||||

outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 2,2 | 2 + 2 | and are both nonzero and are arbitrary | |||||

outside abelian subgroup of maximum order, Jordan blocks of size 1,1,2 | 2 + 1 + 1 | Two subcases: Case 1: , nonzero, arbitrary Case 2: , nonzero, arbitrary | |||||

outside abelian subgroup of maximum order, Jordan blocks of size 2,2 | 2 + 2 | both nonzero arbitrary uniquely determined by other values | |||||

outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order | 3 + 1 | if odd 4 if |
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]
| ||||

outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order | 3 + 1 | if odd 4 if |
Two subcases: Case 1: nonzero, , other entries arbitrary Case 2: nonzero, , other entries arbitrary | ||||

Jordan block of size 4 | 4 | if if |
nonzero arbitrary | ||||

Total (--) | -- | -- | -- | -- | -- |

## Linear representation theory

`Further information: linear representation theory of unitriangular matrix group:UT(4,p)`

### Summary

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

number of conjugacy classes (equals number of irreducible representations over a splitting field) | . See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree four over a finite field |

degrees of irreducible representations | 1 (occurs times), (occurs times), (occurs times) |

sum of squares of degrees of irreducible representations | (equals order of the group) see sum of squares of degrees of irreducible representations equals order of group |

lcm of degrees of irreducible representations |

## GAP implementation

We assume is assigned a prime number value beforehand.

Description | Functions used |
---|---|

SylowSubgroup(GL(4,p),p) |
SylowSubgroup, GL |

SylowSubgroup(SL(4,p),p) |
SylowSubgroup, SL |

SylowSubgroup(PGL(4,p),p) |
SylowSubgroup, PGL |

SylowSubgroup(PSL(4,p),p) |
SylowSubgroup, PSL |