Unitriangular matrix group:UT(3,p)
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Contents
Definition
As a group of matrices
Given a prime , the group
is defined as the unitriangular matrix group of degree three over the prime field
. Explicitly, it has the following form with the usual matrix multiplication:
The analysis given below does not apply to the case . For
, we get the dihedral group:D8, which is studied separately.
As a semidirect product
This group of order can also be described as a semidirect product of the elementary abelian group of order
by the cyclic group of order
, where the generator of the cyclic group of order
acts via the automorphism:
In this case, for instance, we can take the subgroup with as the elementary abelian subgroup of order
and the subgroup with
as the cyclic subgroup of order
.
Definition by presentation
The group can be defined by means of the following presentation:
where 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 correspond to matrices:
In coordinate form
We may define the group as set of triples over the prime field
,
with the multiplication law given by:
.
The matrix corresponding to triple is:
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
.
- 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 | ![]() |
order | ![]() Agrees with general order formula for ![]() ![]() |
conjugacy class size statistics | size 1 (![]() ![]() ![]() |
orbits under automorphism group | Case ![]() Case odd ![]() ![]() ![]() ![]() ![]() ![]() |
number of orbits under automorphism group | 4 if ![]() 3 if ![]() |
order statistics | Case ![]() Case ![]() ![]() ![]() |
exponent | 4 if ![]() ![]() ![]() |
Conjugacy class structure
Note that the characteristic polynomial of all elements in this group is , hence we do not devote a column to the characteristic polynomial.
For reference, we consider matrices of the form:
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 | ![]() |
1 | 1 | 1 | 1 | ![]() |
non-identity element, but central (has Jordan blocks of size one and two respectively) | 2 + 1 | ![]() |
1 | ![]() |
![]() |
![]() |
![]() ![]() |
non-central, has Jordan blocks of size one and two respectively | 2 + 1 | ![]() |
![]() |
![]() |
![]() |
![]() |
![]() ![]() ![]() |
non-central, has Jordan block of size three | 3 | ![]() |
![]() |
![]() |
![]() |
![]() ![]() 4 if ![]() |
both ![]() ![]() |
Total (--) | -- | -- | -- | ![]() |
![]() |
-- | -- |
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 and/or when
. 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 ![]() |
prime-base logarithm of exponent | 1 | the exponent is ![]() ![]() ![]() |
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 | ![]() |
![]() ![]() |
number of subgroups | ![]() ![]() ![]() ![]() |
See subgroup structure of unitriangular matrix group:UT(3,p) |
number of normal subgroups | ![]() |
See subgroup structure of unitriangular matrix group:UT(3,p) |
number of conjugacy classes of subgroups | ![]() ![]() ![]() ![]() |
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 | ![]() |
trivial group | 1 | ![]() |
1 | 1 | 1 | prime-cube order group:U(3,p) | 1 |
center of unitriangular matrix group:UT(3,p) | ![]() ![]() |
group of prime order | ![]() |
![]() |
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 ![]() |
group of prime order | ![]() |
![]() |
![]() |
![]() |
![]() |
-- | 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 | ![]() |
![]() |
![]() |
1 | ![]() |
group of prime order | 1 |
whole group | all elements | unitriangular matrix group:UT(3,p) | ![]() |
1 | 1 | 1 | 1 | trivial group | 0 |
Total (5 rows) | -- | -- | -- | -- | ![]() |
-- | ![]() |
-- | -- |
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 |
Group of prime order | ![]() |
![]() |
![]() |
1 | 1 |
Elementary abelian group of prime-square order | ![]() |
![]() |
![]() |
![]() |
0 |
Prime-cube order group:U3p | ![]() |
1 | 1 | 1 | 1 |
Total | -- | ![]() |
![]() |
![]() |
![]() |
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 | 1 |
![]() |
![]() |
![]() |
![]() |
0 |
![]() |
1 | 1 | 1 | 1 |
Total | ![]() |
![]() |
![]() |
![]() |
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) | ![]() |
degrees of irreducible representations over a splitting field (such as ![]() ![]() |
1 (occurs ![]() ![]() ![]() |
sum of squares of degrees of irreducible representations | ![]() see sum of squares of degrees of irreducible representations equals order of group |
lcm of degrees of irreducible representations | ![]() |
condition for a field (characteristic not equal to ![]() |
The polynomial ![]() For a finite field of size ![]() ![]() |
field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero) | Field ![]() ![]() ![]() ![]() |
unique minimal splitting field (characteristic ![]() |
The field of size ![]() ![]() ![]() ![]() |
degrees of irreducible representations over the rational numbers | 1 (1 time), ![]() ![]() ![]() |
Orbits over a splitting field under the action of the automorphism group | Case ![]() Case odd ![]() ![]() ![]() ![]() number: 4 (for ![]() ![]() |
Orbits over a splitting field under the multiplicative action of one-dimensional representations | Orbit sizes: ![]() ![]() ![]() |
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 ![]() ![]() |
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 , this group is the third group among the groups of order
. Thus, for instance, if
, the group is described using GAP's SmallGroup function as:
SmallGroup(343,3)
Note that we don't need to compute ; we can also write this as:
SmallGroup(7^3,3)
As an extraspecial group
For any prime , we can define this group using GAP's ExtraspecialGroup function as:
ExtraspecialGroup(p^3,'+')
For , it can also be constructed as:
ExtraspecialGroup(p^3,p)
where the argument indicates that it is the extraspecial group of exponent
. For instance, for
:
ExtraspecialGroup(5^3,5)
Endomorphisms
Automorphisms
The automorphisms essentially permute the subgroups of order containing the center, while leaving the center itself unmoved.
Related groups
For any prime , there are (up to isomorphism) two non-abelian groups of order
. One of them is this, and the other is the semidirect product of the cyclic group of order
by a group of order
acting by power maps (with the generator corresponding to multiplication by
).