# Difference between revisions of "Unitriangular matrix group:UT(3,p)"

View other such prime-parametrized groups

## Definition

### As a group of matrices

Given a prime $p$, the group $U_3(p)$ is defined as follows: it is the group of upper triangular matrices with 1s on the diagonal, and entries over $F_p$ (with the group operation being matrix multiplication).

Each such matrix $(a_{ij})$ can be described by the three entries $a_{12}, a_{13}, a_{23}$. The matrix looks like:

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

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

• $c_{12} = a_{12} + b_{12}$
• $c_{13} = a_{13} + b_{13} + a_{12}b_{23}$
• $c_{23} = a_{23} + b_{23}$

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. For further information on the contrast between the case of 2 and of odd primes, refer U3p:odd prime versus two.

### 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 $p$.

## 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 $p^3 - p$

## Subgroups

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:

### Fully characteristic subgroups

The fully characteristic subgroups are precisely the same as the characteristic ones.

### Central factors

The central factors are precisely the same as the characteristic subgroups.

### Direct factors

There are no proper nontrivial direct factors. In other words, the group is directly indecomposable.

## Implementation in GAP

### Group ID

For any prime $p$, this group is the third group amoung the groups of order $p^3$. Thus, for instance, if $p = 7$, the group is described as:

SmallGroup(343,3)

## 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 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 $p+1$).

## Subgroup-defining functions=

### Center

The center of this group is abstractly isomorphic to: group of prime order

### Commutator subgroup

The commutator subgroup of this group is abstractly isomorphic to: group of prime order

### Frattini subgroup

The Frattini subgroup of this group is abstractly isomorphic to: group of prime order

### Socle

The socle of this group is abstractly isomorphic to: group of prime order