SmallGroup(32,31): Difference between revisions

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(Relabeled the presentation to make it more clear which variables are in the normal subgroup.)
 
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==Definition==
==Definition==


This group is defined by the following [[presentation]]:
This group is a [[semidirect product]] <math>(Z_4 \times Z_4) \rtimes Z_2</math> with the following [[presentation]]:


<math>G := \langle a,x,y \mid x^2 = a^4 = y^4 = e, xax^{-1}a^{-1} = y^2, ay = ya, xyx^{-1}y^{-1} = a^2 \rangle</math>
<math>G := \langle a,b,x \mid a^4 = b^4 = x^2 = e, ab = ba, xax^{-1} = a b^2, xbx^{-1} = a^2b \rangle</math>
 
==Arithmetic functions==
 
{{compare and contrast arithmetic functions|order = 32}}
{| class="sortable" border="1"
! Function !! Value !! Similar groups !! Explanation for function value
|-
| [[underlying prime of p-group]] || [[arithmetic function value::underlying prime of p-group;2|2]] || ||
|-
| {{arithmetic function value order|32}} ||
|-
| {{arithmetic function value order p-log etc|5}}
|-
| {{arithmetic function value given order|exponent of a group|4|32}} ||
|-
| {{arithmetic function value given order and p-log|prime-base logarithm of exponent|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|nilpotency class|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|derived length|2|32|5}} ||
|-
| {{arithmetic function value given order and p-log|Frattini length|2|32|5}} ||
|-
| [[Fitting length]] || [[arithmetic function value::Fitting length;1|1]] || || The group is a [[nilpotent group]], hence its [[Fitting length]] is 1. Note that [[prime power order implies nilpotent]], so all groups of the same order have Fitting length 1.
|-
| {{arithmetic function value given order and p-log|minimum size of generating set|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|subgroup rank of a group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|rank of a p-group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|normal rank of a p-group|3|32|5}} ||
|-
| {{arithmetic function value given order and p-log|characteristic rank of a p-group|2|32|5}} || 
|}
 
 
==Group properties==
 
{{compare and contrast group properties|order = 32}}
{| class="sortable" border="1"
! Property !! Satisfied? !! Explanation !! Comment
|-
| {{group properties because p-group}}
|-
| [[dissatisfies property::abelian group]] || No || ||
|}
==GAP implementation==
 
{{GAP ID|32|31}}

Latest revision as of 19:18, 31 January 2021

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]

Definition

This group is a semidirect product with the following presentation:

Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions

Function Value Similar groups Explanation for function value
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 32 groups with same order
prime-base logarithm of order 5 groups with same prime-base logarithm of order
max-length of a group 5 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 5 chief length equals prime-base logarithm of order for group of prime power order
composition length 5 composition length equals prime-base logarithm of order for group of prime power order
exponent of a group 4 groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of exponent 2 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
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
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
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
Fitting length 1 The group is a nilpotent group, hence its Fitting length is 1. Note that prime power order implies nilpotent, so all groups of the same order have Fitting length 1.
minimum size of generating set 3 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 3 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 3 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 3 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 2 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

Want to compare and contrast group properties with other groups of the same order? Check out groups of order 32#Group properties

Property Satisfied? Explanation Comment
group of prime power order Yes
nilpotent group Yes prime power order implies nilpotent
supersolvable group Yes via nilpotent: finite nilpotent implies supersolvable
solvable group Yes via nilpotent: nilpotent implies solvable
abelian group No

GAP implementation

Group ID

This finite group has order 32 and has ID 31 among the groups of order 32 in GAP's SmallGroup library. For context, there are groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,31)

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

gap> G := SmallGroup(32,31);

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

IdGroup(G) = [32,31]

or just do:

IdGroup(G)

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


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