Difference between revisions of "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 $\mathbb{F}_p$.

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 $p$.

Families

1. These groups fall in the more general family $UT(n,p)$ of unipotent upper-triangular matrices, which is the $p$-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 $p$, which is the $p$-Sylow subgroup of $GL(n,q)$.
2. These groups also fall into the general family of extraspecial groups.

Elements

Upto conjugacy

Every element has order $p$.

The conjugacy classes are as follows:

1. The center has order precisely $p$, so there are $p$ elements that form conjugacy classes of size 1. These are, specifically, the elements with $a_{12} = a_{23} = 0$, and they're thus parametrized by their $a_{13}$ entry.
2. For every element outside the center, the centralizer of that element is the subgroup generated by that element and the center, and is hence of order $p^2$. Thus, the conjugacy class of the element is of size $p$. Thus, there is a total of $p^2 - 1$ conjugacy classes of size $p$.

Upto automorphism

There are only three classes of elements upto automorphism:

1. The identity element, which forms a class of size 1
2. The non-identity elements in the center, which form a class of size $p - 1$
3. The non-central elements, which form a class of size $p^3 - p$

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 $p = 2$
number of normal subgroups $\! p + 4$
number of conjugacy classes of subgroups $\! 2p + 5$ for $p \ne 2$, $8$ for $p = 2$

Subgroups

Further information: Subgroup structure of prime-cube order group:U(3,p)

Here is the complete list of subgroups:

1. The trivial subgroup (1)
2. The center, which is a group of order $p$. In matrix terms, this is the subgroup comprising matrices $a_{ij}$ with $a_{12} = a_{23} = 0$. (1)
3. Subgroups of order $p$ generated by non-central elements. These are not normal, and occur in conjugacy classes of size $p$. ($p(p+1)$)
4. Subgroups of order $p^2$ containing the center. These are the inverse images via the quotient map by the center, of subgroups of order $p$ in the inner automorphism group. ($p + 1$)
5. The whole group. (1)

Normal subgroups

The subgroups in (1), (2), (4) and (5) above are normal.

Characteristic subgroups

The subgroups in (1), (2) and (5) above are normal. In other words, there are only three characteristic subgroups. Some notable facts:

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 $p+1$).