Unitriangular matrix group:UT(3,3): Difference between revisions

Revision as of 00:50, 4 July 2010

This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
View a complete list of particular groups (this is a very huge list!)[SHOW MORE]

VIEW FACTS ABOUT THIS GROUP: All factsGroup property satisfactionsSubgroup property satisfactionsGroup property dissatisfactionsSubgroup property satisfactions
VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

Definition

This group is defined in the following equivalent ways:

It is the unique (up to isomorphism) non-abelian group of order $27$ and exponent $3$ .
It is the upper-triangular unipotent matrix group: the group of $3\times 3$ matrices over the field of three elements.
It is the inner automorphism group of wreath product of groups of order p for $p=3$ .
It is the Burnside group $B(2,3)$ : the quotient of the free group of rank two by the subgroup generated by all cubes in the group.

Families

Prime-cube order group:U(3,p): For an odd prime $p$ , this is the unique non-abelian group of order $p^{3}$ and exponent $p$ . It is the group of unipotent upper-triangular $3\times 3$ matrices over the field of three elements.
Burnside groups: This group is $B(2,3)$ . In general, the Burnside groups $B(m,3)$ are all finite.

Arithmetic functions

Function	Value	Similar groups
order (number of elements, equivalently, cardinality or size of underlying set)	27	groups with same order
underlying prime of p-group	3
prime-base logarithm of order	3
exponent of a group	3	groups with same order and exponent of a group \| groups with same exponent of a group
prime-base logarithm of exponent	1	groups with same order and prime-base logarithm of exponent \| groups with same prime-base logarithm of order and prime-base logarithm of exponent \| groups with same prime-base logarithm of exponent
Frattini length	2	groups with same order and Frattini length \| groups with same prime-base logarithm of order and Frattini length \| groups with same Frattini length
derived length	2	groups with same order and derived length \| groups with same prime-base logarithm of order and derived length \| groups with same derived length
nilpotency class	2	groups with same order and nilpotency class \| groups with same prime-base logarithm of order and nilpotency class \| groups with same nilpotency class
minimum size of generating set	2	groups with same order and minimum size of generating set \| groups with same prime-base logarithm of order and minimum size of generating set \| groups with same minimum size of generating set
subgroup rank of a group	2	groups with same order and subgroup rank of a group \| groups with same prime-base logarithm of order and subgroup rank of a group \| groups with same subgroup rank of a group
rank of a p-group	2	groups with same order and rank of a p-group \| groups with same prime-base logarithm of order and rank of a p-group \| groups with same rank of a p-group
normal rank of a p-group	2	groups with same order and normal rank of a p-group \| groups with same prime-base logarithm of order and normal rank of a p-group \| groups with same normal rank of a p-group
characteristic rank of a p-group	1	groups with same order and characteristic rank of a p-group \| groups with same prime-base logarithm of order and characteristic rank of a p-group \| groups with same characteristic rank of a p-group

Group properties

Property	Satisfied	Explanation
abelian group	No
group of prime power order	Yes
nilpotent group	Yes	prime power order implies nilpotent
solvable group	Yes
extraspecial group	Yes
Frattini-in-center group	Yes

Other associated constructs

Associated construct	Isomorphism class	Comment
Lazard Lie ring	upper-triangular nilpotent Lie ring:u(3,3)

GAP implementation

Group ID

This finite group has order 27 and has ID 3 among the groups of order 27 in GAP's SmallGroup library. For context, there are groups of order 27. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(27,3)

For instance, we can use the following assignment in GAP to create the group and name it $G$ :

gap> G := SmallGroup(27,3);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [27,3]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

@@ Line 18: / Line 18: @@
 {| class="wikitable" border="1"
-! Function !! Value !! Explanation
+! Function !! Value !! Similar groups !! Explanation for function value
 |-
-| [[order of a group|order]] || [[arithmetic function value::order of a group;27|27]] ||
+| {{arithmetic function value order|27}} ||
 |-
-| [[exponent of a group|exponent]] || [[arithmetic function value::exponent of a group;3|3]] ||
+| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;3|3]] ||
 |-
-| [[Frattini length]] || [[arithmetic function value::Frattini length;2|2]] ||
+| [[prime-base logarithm of order]] || [[arithmetic function value::prime-base logarithm of order;3|3]] || ||
 |-
-| [[Fitting length]] || [[arithmetic function value::Fitting length;1|1]] ||
+| {{arithmetic function value given order|exponent of a group|3|27}} ||
 |-
-| [[derived length]] || [[arithmetic function value::derived length;2|2]] ||
+| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|1|27|3}} ||
 |-
-| [[nilpotency class]] || [[arithmetic function value::nilpotency class;2|2]] ||
+| {{arithmetic function value given order and p-log|Frattini length|2|27|3}} ||
 |-
-| [[minimum size of generating set]] || [[arithmetic function value::minimum size of generating set;2|2]] ||
+| {{arithmetic function value given order and p-log|derived length|2|27|3}} ||
 |-
-| [[subgroup rank of a group|subgroup rank]] || [[arithmetic function value::subgroup rank of a group;2|2]] ||
+| {{arithmetic function value given order and p-log|nilpotency class|2|27|3}}
 |-
-| [[rank of a p-group|rank as p-group]] || [[arithmetic function value::rank of a p-group;2|2]] ||
+| {{arithmetic function value given order and p-log|minimum size of generating set|2|27|3}} ||
 |-
-| [[normal rank of a p-group|normal rank as p-group]] || [[arithmetic function value::normal rank of a p-group;2|2]] ||
+| {{arithmetic function value given order and p-log|subgroup rank of a group|2|27|3}} ||
 |-
-| [[characteristic rank of a p-group|characteristic rank as p-group]] || [[arithmetic function value::characteristic rank of a p-group;1|1]]
+| {{arithmetic function value given order and p-log|rank of a p-group|2|27|3}} ||
+|-
+| {{arithmetic function value given order and p-log|normal rank of a p-group|2|27|3}} ||
+|-
+| {{arithmetic function value given order and p-log|characteristic rank of a p-group|1|27|3}}
 |}