Unitriangular matrix group:UT(3,p)

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Definition

As a group of matrices

Given a prime $p$, the group $UT(3,p)$ is defined as the unitriangular matrix group of degree three over the prime field $\mathbb{F}_p$. Explicitly, it has the following form with the usual matrix multiplication:

$\left \{ \begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix} \mid a_{12},a_{13},a_{23} \in \mathbb{F}_p \right \}$

The analysis given below does not apply to the case $p = 2$. For $p = 2$, we get the dihedral group:D8, which is studied separately.

As a semidirect product

This group of order $p^3$ can also be described as a semidirect product of the elementary abelian group of order $p^2$ by the cyclic group of order $p$, where the generator of the cyclic group of order $p$ acts via the automorphism:

$(a,b) \mapsto (a,a+b)$

In this case, for instance, we can take the subgroup with $a_{12} = 0$ as the elementary abelian subgroup of order $p^2$ and the subgroup with $a_{23} = a_{13} = 0$ as the cyclic subgroup of order $p$.

Definition by presentation

The group can be defined by means of the following presentation:

$\langle x,y,z \mid [x,y] = z, xz = zx, yz = zy, x^p = y^p = z^p = e \rangle$

where $e$ denotes the identity element.

These commutation relation resembles Heisenberg's commuatation relations in quantum mechanics and so the group is sometimes called a finite Heisenberg group. Generators $x,y,z$ correspond to matrices:

$x=\begin{pmatrix} 1 & 1 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix},\ \ y=\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 1\\ 0 & 0 & 1\\ \end{pmatrix},\ \ z=\begin{pmatrix} 1 & 0 & 1\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}$

In coordinate form

We may define the group as set of triples $(a_{12},a_{13},a_{23})$ over the prime field $\mathbb{F}_p$, with the multiplication law given by:

$(a_{12},a_{13},a_{23}) (b_{12},b_{13},b_{23}) = (a_{12} + b_{12},a_{13} + b_{13} + a_{12}b_{23}, a_{23} + b_{23}), ~~~~ (a_{12},a_{13},a_{23})^{-1} = (-a_{12}, -a_{13} + a_{12}a_{23}, a_{23})$.

The matrix corresponding to triple $(a_{12},a_{13},a_{23})$ is:

$\begin{pmatrix} 1 & a_{12} & a_{13}\\ 0 & 1 & a_{23}\\ 0 & 0 & 1\\ \end{pmatrix}$

Families

1. These groups fall in the more general family $UT(n,p)$ of unitriangular matrix groups. The unitriangular matrix group $UT(n,p)$ can be described as the group of unipotent upper-triangular matrices in $GL(n,p)$, which is also a $p$-Sylow subgroup of the general linear group $GL(n,p)$. This further can be generalized to $UT(n,q)$ where $q$ is the power of a prime $p$. $UT(n,q)$ is the $p$-Sylow subgroup of $GL(n,q)$.
2. These groups also fall into the general family of extraspecial groups.

Elements

Further information: element structure of unitriangular matrix group:UT(3,p)

Summary

Item	Value
number of conjugacy classes	$p^2 + p - 1$
order	$p^3$ Agrees with general order formula for $UT(n,q)$: $q^{n(n-1)/2} = p^{(3)(2)/2} = p^3$
conjugacy class size statistics	size 1 ($p$ times), size $p$ ($p^2 - 1$ times)
orbits under automorphism group	Case $p = 2$: size 1 (1 conjugacy class of size 1), size 1 (1 conjugacy class of size 1), size 2 (1 conjugacy class of size 2), size 4 (2 conjugacy classes of size 2 each) Case odd $p$: size 1 (1 conjugacy class of size 1), size $p - 1$ ($p - 1$ conjugacy classes of size 1 each), size $p^3 - p$ ($p^2 - 1$ conjugacy classes of size $p$ each)
number of orbits under automorphism group	4 if $p = 2$ 3 if $p$ is odd
order statistics	Case $p = 2$: order 1 (1 element), order 2 (5 elements), order 4 (2 elements) Case $p$ odd: order 1 (1 element), order $p$ ($p^3 - 1$ elements)
exponent	4 if $p = 2$ $p$ if $p$ odd

Conjugacy class structure

Note that the characteristic polynomial of all elements in this group is $(t - 1)^3$, hence we do not devote a column to the characteristic polynomial.

For reference, we consider matrices of the form:

$\begin{pmatrix} 1 & a_{12} & a_{13} \\ 0 & 1 & a_{23} \\ 0 & 0 & 1 \\\end{pmatrix}$

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
identity element	1 + 1 + 1 + 1	$t - 1$	1	1	1	1	$a_{12} = a_{13} = a_{23} = 0$
non-identity element, but central (has Jordan blocks of size one and two respectively)	2 + 1	$(t - 1)^2$	1	$p - 1$	$p - 1$	$p$	$a_{12} = a_{23} = 0$, $a_{13} \ne 0$
non-central, has Jordan blocks of size one and two respectively	2 + 1	$(t - 1)^2$	$p$	$2(p - 1)$	$2p(p - 1)$	$p$	$a_{12}a_{23} = 0$, but not both $a_{12}$ and $a_{23}$ are zero
non-central, has Jordan block of size three	3	$(t - 1)^3$	$p$	$(p - 1)^2$	$p(p - 1)^2$	$p$ if $p$ odd 4 if $p = 2$	both $a_{12}$ and $a_{23}$ are nonzero
Total (--)	--	--	--	$p^2 + p - 1$	$p^3$	--	--

Arithmetic functions

Compare and contrast arithmetic function values with other groups of prime-cube order at Groups of prime-cube order#Arithmetic functions

For some of these, the function values are different when $p = 2$ and/or when $p = 3$. These are clearly indicated below.

Arithmetic functions taking values between 0 and 3

Function	Value	Explanation
prime-base logarithm of order	3	the order is $p^3$
prime-base logarithm of exponent	1	the exponent is $p^1$. Exception when $p = 2$, where the exponent is $2^2 = 4$.
nilpotency class	2
derived length	2
Frattini length	2
minimum size of generating set	2
subgroup rank	2
rank as p-group	2
normal rank as p-group	2
characteristic rank as p-group	1

Arithmetic functions of a counting nature

Function	Value	Explanation
number of conjugacy classes	$\! p^2 + p - 1$	$p$ elements in the center, and each other conjugacy class has size $p$
number of subgroups	$\! p^2 + 2p + 4$ when $p \ne 2$, $10$ when $p = 2$	See subgroup structure of unitriangular matrix group:UT(3,p)
number of normal subgroups	$\! p + 4$	See subgroup structure of unitriangular matrix group:UT(3,p)
number of conjugacy classes of subgroups	$\! 2p + 5$ for $p \ne 2$, $8$ for $p = 2$	See subgroup structure of unitriangular matrix group:UT(3,p)

Subgroups

Further information: Subgroup structure of unitriangular matrix group:UT(3,p)

Table classifying subgroups up to automorphisms

Automorphism class of subgroups	Representative	Isomorphism class	Order of subgroups	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	$\left \{ \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}\right \}$	trivial group	1	$p^3$	1	1	1	prime-cube order group:U(3,p)	1
center of unitriangular matrix group:UT(3,p)	$\langle \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\\end{pmatrix}\rangle$; equivalently, given by $a_{12} = a_{23} = 0$.	group of prime order	$p$	$p^2$	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 $a_{12}, a_{23}$ nonzero	group of prime order	$p$	$p^2$	$p + 1$	$p$	$p(p + 1)$	--	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	$p^2$	$p$	$p + 1$	1	$p + 1$	group of prime order	1
whole group	all elements	unitriangular matrix group:UT(3,p)	$p^3$	1	1	1	1	trivial group	0
Total (5 rows)	--	--	--	--	$2p + 5$	--	$p^2 + 2p + 4$	--	--

Tables classifying isomorphism types of subgroups

Group name	GAP ID	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
Trivial group	$(1,1)$	1	1	1	1
Group of prime order	$(p,1)$	$p^2 + p + 1$	$p + 2$	1	1
Elementary abelian group of prime-square order	$(p^2,2)$	$p + 1$	$p + 1$	$p + 1$	0
Prime-cube order group:U3p	$(p^3,3)$	1	1	1	1
Total	--	$p^2 + 2p + 4$	$2p + 5$	$p + 4$	$3$

Table listing number of subgroups by order

Group order	Occurrences as subgroup	Conjugacy classes of occurrence as subgroup	Occurrences as normal subgroup	Occurrences as characteristic subgroup
$1$	1	1	1	1
$p$	$p^2 + p + 1$	$p + 2$	1	1
$p^2$	$p + 1$	$p + 1$	$p + 1$	0
$p^3$	1	1	1	1
Total	$p^2 + 2p + 4$	$2p + 5$	$p + 4$	$3$

Linear representation theory

Further information: linear representation theory of unitriangular matrix group:UT(3,p)

Item	Value
number of conjugacy classes (equals number of irreducible representations over a splitting field)	$p^2 + p - 1$. See number of irreducible representations equals number of conjugacy classes, element structure of unitriangular matrix group of degree three over a finite field
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$)	1 (occurs $p^2$ times), $p$ (occurs $p - 1$ times)
sum of squares of degrees of irreducible representations	$p^3$ (equals order of the group) see sum of squares of degrees of irreducible representations equals order of group
lcm of degrees of irreducible representations	$p$
condition for a field (characteristic not equal to $p$) to be a splitting field	The polynomial $x^p - 1$ should split completely. For a finite field of size $q$, this is equivalent to $q \equiv 1 \pmod p$.
field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero)	Field $\mathbb{Q}(\zeta)$ where $\zeta$ is a primitive $p^{th}$ root of unity. This is a degree $p - 1$ extension of the rationals.
unique minimal splitting field (characteristic $c \ne 0,p$)	The field of size $c^r$ where $r$ is the order of $c$ mod $p$.
degrees of irreducible representations over the rational numbers	1 (1 time), $p - 1$ ($p + 1$ times), $p(p - 1)$ (1 time)
Orbits over a splitting field under the action of the automorphism group	Case $p = 2$: Orbit sizes: 1 (degree 1 representation), 1 (degree 1 representation), 2 (degree 1 representations), 1 (degree 2 representation) Case odd $p$: Orbit sizes: 1 (degree 1 representation), $p^2 - 1$ (degree 1 representations), $p - 1$ (degree $p$ representations) number: 4 (for $p = 2$), 3 (for odd $p$)
Orbits over a splitting field under the multiplicative action of one-dimensional representations	Orbit sizes: $p^2$ (degree 1 representations), and $p - 1$ orbits of size 1 (degree $p$ representations)

Subgroup-defining functions

Subgroup-defining function	Subgroup type in list	Isomorphism class	Comment
Center	(2)	Group of prime order
Commutator subgroup	(2)	Group of prime order
Frattini subgroup	(2)	Group of prime order	The $p + 1$ maximal subgroups of order $p^2$ intersect here.
Socle	(2)	Group of prime order	This subgroup is the unique minimal normal subgroup, i.e.,the monolith, and the group is monolithic. Also, minimal normal implies central in nilpotent.

Quotient-defining function

Quotient-defining function	Isomorphism class	Comment
Inner automorphism group	Elementary abelian group of prime-square order	It is the quotient by the center, which is of prime order.
Abelianization	Elementary abelian group of prime-square order	It is the quotient by the commutator subgroup, which is of prime order.
Frattini quotient	Elementary abelian group of prime-square order	It is the quotient by the Frattini subgroup, which is of prime order.

GAP implementation

GAP ID

For any prime $p$, this group is the third group among the groups of order $p^3$. Thus, for instance, if $p = 7$, the group is described using GAP's SmallGroup function as:

SmallGroup(343,3)

Note that we don't need to compute $p^3$; we can also write this as:

SmallGroup(7^3,3)

As an extraspecial group

For any prime $p$, we can define this group using GAP's ExtraspecialGroup function as:

ExtraspecialGroup(p^3,'+')

For $p \ne 2$, it can also be constructed as:

ExtraspecialGroup(p^3,p)

where the argument $p$ indicates that it is the extraspecial group of exponent $p$. For instance, for $p = 5$:

ExtraspecialGroup(5^3,5)

Endomorphisms

Automorphisms

The automorphisms essentially permute the subgroups of order $p^2$ containing the center, while leaving the center itself unmoved.

Related groups

For any prime $p$, there are (up to isomorphism) two non-abelian groups of order $p^3$. One of them is this, and the other is the semidirect product of the cyclic group of order $p^2$ by a group of order $p$ acting by power maps (with the generator corresponding to multiplication by $p+1$).

External links

• Wikipedia page