View source for Subgroup structure of unitriangular matrix group:UT(3,p)

{{group family-specific information|
group family = unitriangular matrix group:UT(3,p)|
information type = subgroup structure|
connective = of}}

Let <math>p</math> be an odd prime.

The group <math>G = U(3,p)</math> is the group of unipotent upper-triangular matrices over the prime field of <math>p</math> elements, and is also the unique (up to isomorphism) non-abelian group of order <math>p^3</math> and exponent <math>p</math>. It is defined by the presentation:

<math>G := \langle g,h,k \mid g^p = h^p = k^p = e, gh = hg, gk = kg, khk^{-1} = g^{-1}h \rangle</math>.

In the matrix description, each matrix <math>(a_{ij})</math> can be described by the three entries <math>a_{12}, a_{13}, a_{23}</math>. The matrix looks like:

<math>\begin{pmatrix}
1 & a_{12} & a_{13} \\
0 & 1 & a_{23}\\
0 & 0 & 1\end{pmatrix}</math>

The multiplication  of matrices <math>A = (a_{ij})</math> and <math>B = (b_{ij})</math> gives the matrix <math>C = (c_{ij})</math> where:

* <math>c_{12} = a_{12} + b_{12}</math>
* <math>c_{13} = a_{13} + b_{13} + a_{12}b_{23}</math>
* <math>c_{23} = a_{23} + b_{23}</math>

With the matrix description, we can set <math>g</math> as the matrix with <math>a_{13} = 1</math> and the other two entries zero, <math>h</math> as the matrix with <math>a_{12} = 1</math> and the other two entries zero, and <math>k</math> as the matrix with <math>a_{23} = 1</math> and the other two entries zero.

The group has the following subgroups:

==Tables for quick information==

{{finite p-group subgroup structure facts to check against}}
<section begin="summary"/>
===Table classifying subgroups up to automorphisms===

{| class="sortable" border="1"
! Automorphism class of subgroups !! Representative !! Isomorphism class !! [[Order of a group|Order]] of subgroups !! [[Index of a subgroup|Index]] of subgroups !! Number of conjugacy classes !! Size of each conjugacy class !! Number of subgroups !! Isomorphism class of quotient (if exists) !! [[Subnormal depth]] (if subnormal)
|-
| trivial subgroup || <math>\left \{ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}\right \}</math> || [[trivial group]] || 1 || <math>p^3</math> || 1 || 1 || 1 || [[prime-cube order group:U(3,p)]] || 1
|-
| [[center of unitriangular matrix group:UT(3,p)]] || <math>\langle \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}\rangle </math>; equivalently, given by <math>a_{12} = a_{23} = 0</math>. || [[group of prime order]] || <math>p</math> || <math>p^2</math> || 1 || 1 || 1 || [[elementary abelian group of prime-square order]] || 1
|-
| [[non-central subgroups of prime order in unitriangular matrix group:UT(3,p)]] || Subgroup generated by any element with at least one of the entries <math>a_{12}, a_{23}</math> nonzero || [[group of prime order]] || <math>p</math> || <math>p^2</math> || <math>p + 1</math> || <math>p</math> || <math>p(p + 1)</math> || -- || 2
|-
| [[elementary abelian subgroups of prime-square order in unitriangular matrix group:UT(3,p)]] || join of center and any non-central subgroup of prime order || [[elementary abelian group of prime-square order]] || <math>p^2</math> || <math>p</math> || <math>p + 1</math> || 1 || <math>p + 1</math> || [[group of prime order]] || 1
|-
| whole group || all elements || [[unitriangular matrix group:UT(3,p)]] || <math>p^3</math> || 1 || 1 || 1 || 1 || [[trivial group]] || 0
|-
! Total (5 rows) || -- || -- || -- || -- || <math>2p + 5</math> || -- || <math>p^2 + 2p + 4</math> || -- || --
|}

===Tables classifying isomorphism types of subgroups===

{| class="sortable" border="1"
! Group name !! GAP ID !! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
|-
| [[Trivial group]] || <math>(1,1)</math> || 1 || 1 || 1 || 1
|-
| [[Group of prime order]] || <math>(p,1)</math> || <math>p^2 + p + 1</math> || <math>p + 2</math> || 1 || 1
|-
| [[Elementary abelian group of prime-square order]] || <math>(p^2,2)</math> || <math>p + 1</math> || <math>p + 1</math> || <math>p + 1</math> || 0
|-
| [[Prime-cube order group:U3p]] || <math>(p^3,3)</math> || 1 || 1 || 1 || 1
|-
! Total || -- || <math>p^2 + 2p + 4</math> || <math>2p + 5</math> || <math>p + 4</math> || <math>3</math>
|}

===Table listing number of subgroups by order===

{| class="sortable" border="1"
! Group order!! Occurrences as subgroup !! Conjugacy classes of occurrence as subgroup !! Occurrences as normal subgroup !! Occurrences as characteristic subgroup
|-
| <math>1</math> || 1 || 1 || 1 || 1
|-
| <math>p</math> || <math>p^2 + p + 1</math> || <math>p + 2</math> || 1 || 1
|-
| <math>p^2</math> || <math>p + 1</math> || <math>p + 1</math> || <math>p + 1</math> || 0
|-
| <math>p^3</math> || 1 || 1 || 1 || 1
|-
! Total|| <math>p^2 + 2p + 4</math> || <math>2p + 5</math> || <math>p + 4</math> || <math>3</math>
|}
<section end="summary"/>