SmallGroup(32,2)

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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
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Definition

This group can be defined by the following presentation:

G := \langle a_1,a_2,a_3 \mid a_1^4 = a_2^4 = a_3^2 = e,  [a_1,a_2] = a_3, [a_1,a_3] = [a_2,a_3] = e \rangle

where e denotes the identity element and [ , ] stands for the commutator of two elements (the isomorphism type of the group is independent of the choice of commutator map).

See the section #Description by presentation for how to construct the group using this presentation in GAP.

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 GAP verification (set G := SmallGroup(32,2);) -- see more at #GAP implementation
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 32 groups with same order Order(G); using 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 prime-base logarithm of order and exponent of a group | groups with same exponent of a group Exponent(G); using Exponent.
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
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 FrattiniLength(G); using FrattiniLength.
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 NilpotencyClassOfGroup(G); using NilpotencyClassOfGroup.
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 DerivedLength(G); using DerivedLength.
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 Rank(G); using Rank.
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 RankAsPGroup(G); using RankAsPGroup.
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 NormalRank(G); using NormalRank.
characteristic rank of a p-group 3 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 CharacteristicRank(G); using CharacteristicRank.

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
T-group No
monolithic group No
one-headed group No
group of nilpotency class two Yes
metabelian group Yes
finite group that is 1-isomorphic to an abelian group Yes via cocycle skew reversal generalization of Baer correspondence

GAP implementation

Group ID

This finite group has order 32 and has ID 2 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(32,2)

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

gap> G := SmallGroup(32,2);

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

IdGroup(G) = [32,2]

or just do:

IdGroup(G)

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


Description by presentation

gap> F := FreeGroup(3);
<free group on the generators [ f1, f2, f3 ]>
gap> G := F/[F.1^4,F.2^4,F.3^2,Comm(F.1,F.2)*F.3^(-1),Comm(F.1,F.3),Comm(F.2,F.3)];
<fp group on the generators [ f1, f2, f3 ]>
gap> IdGroup(G);
[ 32, 2 ]

GAP verification of function values and group properties

Below is a GAP implementation verifying the various function values and group properties as stated in this page. Before beginning, set G := SmallGroup(32,2); or set it to this group in any other manner:

gap> Order(G);
32
gap> Exponent(G);
4
gap> NilpotencyClassOfGroup(G);
2
gap> DerivedLength(G);
2
gap> FrattiniLength(G);
2
gap> Rank(G);
2
gap> RankAsPGroup(G);
3
gap> NormalRank(G);
3
gap> CharacteristicRank(G);
3