Element structure of unitriangular matrix group of degree four over a finite field
This article gives specific information, namely, element structure, about a family of groups, namely: unitriangular matrix group of degree four.
View element structure of group families  View other specific information about unitriangular matrix group of degree four
This article describes in detail the element structure of the unitriangular matrix group of degree four over a finite field. We denote the field size by , the field characteristic by , and the value by . Further, we denote the group as .
Summary
Item  Value 

number of conjugacy classes  equals number of irreducible representations. See number of irreducible representations equals number of conjugacy classes, linear representation theory of unitriangular matrix group of degree four over a finite field 
order  Follows from the general formula, order of is 
conjugacy class size statistics  1 ( times), ( times), ( times), ( times) 
order statistics  Case : order 1 (1 element), order 2 ( elements), order 4 ( elements) Case : order 1 (1 element), order 3 ( elements), order 9 ( elements) Case : order 1 (1 element), order ( elements) 
exponent  if if The exponent depends only on , not on . 
Conjugacy class structure
Number of conjugacy classes
The general theory says that number of conjugacy classes in unitriangular matrix group of fixed degree over a finite field is polynomial function of field size, where the degree of the polynomial is one less than the degree of matrices. Thus, we expect that the number of conjugacy classes is a polynomial function of of degree 4  1 = 3. Indeed, this is the case, and the explicit polynomial is .
Conjugacy class structure in the unitriangular matrix group
For the rightmost column for the type of matrix, we use the (row number, column number) notation for matrix entries. Explicitly, the matrix under consideration is:
The subgroups mentioned in the table below are:
Subgroup  Visual description  Condition  Order 

center  
derived subgroup  
unique abelian subgroup of maximum order 
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 (constraints on ) 

identity element  1 + 1 + 1 + 1  1  1  1  1  all the are zero  
nonidentity element, but central (has Jordan blocks of size 1,1,2 respectively)  2 + 1 + 1  1  , all the others are zero  
noncentral but in derived subgroup, has Jordan blocks of size 1,1,2  2 + 1 + 1  Among and , exactly one of them is nonzero. may be zero or nonzero  
noncentral but in derived subgroup, Jordan blocks of size 2,2  2 + 2  Both and are nonzero. may be zero or nonzero  
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 1,1,2  2 + 1 + 1  is nonzero and are arbitrary  
outside derived subgroup, inside unique abelian subgroup of maximum order, with Jordan blocks of size 2,2  2 + 2  and are both nonzero and are arbitrary  
outside abelian subgroup of maximum order, Jordan blocks of size 1,1,2  2 + 1 + 1  Two subcases: Case 1: , nonzero, arbitrary Case 2: , nonzero, arbitrary  
outside abelian subgroup of maximum order, Jordan blocks of size 2,2  2 + 2  both nonzero arbitrary uniquely determined by other values  
outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order  3 + 1  if odd 4 if 
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outside abelian subgroup of maximum order, Jordan blocks of size 1,3, with centralizer of order  3 + 1  if odd 4 if 
Two subcases: Case 1: nonzero, , other entries arbitrary Case 2: nonzero, , other entries arbitrary  
Jordan block of size 4  4  if if 
nonzero arbitrary  
Total ()           
Grouping by conjugacy class sizes
This follows by computing from the table in the previous section.
Conjugacy class size  Total number of conjugacy classes of this size  Total number of elements  Cumulative number of conjugacy classes  Cumulative number of elements 

1  
(total)  (total) 
Conjugacy classes with respect to the general linear group
If we consider the action of the general linear group by conjugation, then there is considerable fusion of conjugacy classes. Specifically, there are only five equivalence classes, corresponding to the set of unordered integer partitions of 4 describing the possible Jordan block decompositions.
Below is a summary of the information:
Jordan block size decomposition (partition of 4)  Number of conjugacy classes of size 1 Number of elements in these 
Number of conjugacy classes of size Number of elements in these 
Number of conjugacy classes of size Number of elements in these 
Number of conjugacy classes of size Number of elements in these 
Total (number of conjugacy classes, elements) 

1 + 1 + 1 + 1  1 1 
0 0 
0 0 
0 0 
1 1 
2 + 1 + 1  


0 0 

2 + 2  0 0 


0 0 

3 + 1  0 0 
0 0 



4  0 0 
0 0 
0 0 


Total ()  




Order statistics
The order statistics can be computed from the information in the #Conjugacy class structure section. The computations are below:
Case
Order  List of conjugacy class sizes of elements with that order  Total number of conjugacy classes  Total number of elements 

1  size 1 (1 time)  1  1 
(i.e., 2)  size 1 ( times), size ( times), size ( times)  
(i.e., 4)  size ( times), size ( times)  
Total ()   
Case
Order  List of conjugacy class sizes of elements with that order  Total number of conjugacy classes  Total number of elements 

1  size 1 (1 time)  1  1 
(i.e., 3)  size 1 ( times), size ( times), size ( times), size ( times)  
(i.e., 9)  size ( times)  
Total ()   
Case
In this case, all the nonidentity elements have order .
Order  List of conjugacy class sizes of elements with that order  Total number of conjugacy classes  Total number of elements 

1  size 1 (1 time)  1  1 
size 1 ( times), size ( times), size ( times), size ( times)  
Total ()   