Difference between revisions of "Unitriangular matrix group:UT(3,p)"
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===As a semidirect product===
===As a semidirect product===
Revision as of 15:40, 18 September 2012
This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.
View other such prime-parametrized groups
- 1 Definition
- 2 Families
- 3 Elements
- 4 Arithmetic functions
- 5 Subgroups
- 6 Linear representation theory
- 7 Endomorphisms
- 8 GAP implementation
- 9 External links
Note that the case , where the group becomes dihedral group:D8, behaves somewhat differently from the general case. We note on the page all the places where the discussion does not apply to .
As a group of matrices
The multiplication of matrices and gives the matrix where:
The identity element is the identity matrix.
The inverse of a matrix is the matrix where:
Note that all addition and multiplication in these definitions is happening over the field .
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:
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:
Note that in the above presentation, the generator is redundant, and the presentation can thus be rewritten as a presentation with only two generators and .
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 , with the following action. Denote the base of the semidirect product as ordered pairs of elements from . The action of the generator of the acting group is as follows:
In this case, for instance, we can take the subgroup with as the elementary abelian subgroup of order , i.e., the elementary abelian subgroup of order is the subgroup:
The acting subgroup of order can be taken as the subgroup with , i.e., the subgroup:
- 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. For any number of the form , there are two extraspecial groups of that order: an extraspecial group of "+" type and an extraspecial group of "-" type. is an extraspecial group of order and "+" type. The other type of extraspecial group of order , i.e., the extraspecial group of order and "-" type, is semidirect product of cyclic group of prime-square order and cyclic group of prime order.
Further information: element structure of unitriangular matrix group:UT(3,p)
|number of conjugacy classes|
Agrees with general order formula for :
|conjugacy class size statistics||size 1 ( times), size ( times)|
|orbits under automorphism group|| Case : 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 : size 1 (1 conjugacy class of size 1), size ( conjugacy classes of size 1 each), size ( conjugacy classes of size each)
|number of orbits under automorphism group|| 4 if |
3 if is odd
|order statistics|| Case : order 1 (1 element), order 2 (5 elements), order 4 (2 elements)|
Case odd: order 1 (1 element), order ( elements)
|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||, but not both and are zero|
|non-central, has Jordan block of size three||3|| if odd
|both and are nonzero|
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
|prime-base logarithm of order||3||the order is|
|prime-base logarithm of exponent||1||the exponent is . Exception when , where the exponent is .|
|minimum size of generating set||2|
|rank as p-group||2|
|normal rank as p-group||2|
|characteristic rank as p-group||1|
Arithmetic functions of a counting nature
|number of conjugacy classes||elements in the center, and each other conjugacy class has size|
|number of subgroups||when , when||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||for , for||See subgroup structure of unitriangular matrix group:UT(3,p)|
Further information: Subgroup structure of unitriangular matrix group:UT(3,p)
Note that the analysis here specifically does not apply to the case . For , see subgroup structure of dihedral group:D8.
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)||; equivalently, given by .||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 nonzero||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|
|Group of prime order||1||1|
|Elementary abelian group of prime-square order||0|
|Prime-cube order group:U3p||1||1||1||1|
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|
Linear representation theory
Further information: linear representation theory of unitriangular matrix group:UT(3,p)
|number of conjugacy classes (equals number of irreducible representations over a splitting field)||. 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 or )||1 (occurs times), (occurs times)|
|sum of squares of degrees of irreducible representations|| (equals order of the group)|
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 ) to be a splitting field|| The polynomial should split completely.|
For a finite field of size , this is equivalent to .
|field generated by character values, which in this case also coincides with the unique minimal splitting field (characteristic zero)||Field where is a primitive root of unity. This is a degree extension of the rationals.|
|unique minimal splitting field (characteristic )||The field of size where is the order of mod .|
|degrees of irreducible representations over the rational numbers||1 (1 time), ( times), (1 time)|
|Orbits over a splitting field under the action of the automorphism group|| Case : Orbit sizes: 1 (degree 1 representation), 1 (degree 1 representation), 2 (degree 1 representations), 1 (degree 2 representation)|
Case odd : Orbit sizes: 1 (degree 1 representation), (degree 1 representations), (degree representations)
number: 4 (for ), 3 (for odd )
|Orbits over a splitting field under the multiplicative action of one-dimensional representations||Orbit sizes: (degree 1 representations), and orbits of size 1 (degree representations)|
The automorphisms essentially permute the subgroups of order containing the center, while leaving the center itself unmoved.
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:
Note that we don't need to compute ; we can also write this as:
As an extraspecial group
For any prime , we can define this group using GAP's ExtraspecialGroup function as:
For , it can also be constructed as:
where the argument indicates that it is the extraspecial group of exponent . For instance, for :