Burnside group:B(2,4)
From Groupprops
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 defined as the Burnside group with 2 generators and exponent 4. Explicitly, it is the quotient group of free group:F2 by relations that say that the fourth power of every element is the identity element. Note that this presentation would involve infinitely many relations, but since it turns out that the group is finite, we can use a finite presentation instead.
Arithmetic functions
Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 4096#Arithmetic functions
GAP implementation
The group is too big to have a GAP ID, but it can be constructed by imposing fourth power relations:
gap> F := FreeGroup(2);; gap> R1 := [F.1,F.2,(F.1*F.2),Comm(F.1,F.2),(F.1*F.2^2),(F.1^(-1)*F.2),(F.1^(-1)*F.2^2),Comm(F.1,Comm(F.1,F.2)),Comm(F.1,F.2^2),Comm(F.1^(-1),F.2)];; gap> R2 := [F.1*F.2*F.1,F.2*F.1*F.2,F.1*F.2*F.1^2,Comm(F.1,F.1*F.2*F.1)];; gap> R := Union(R1,R2);; gap> S := List(R,x -> x^4);; gap> K := F/S;; gap> U := Set(List(Set(K),x -> x^4));; gap> G := K/(Group(U));;
We can then check the essential facts about the group:
gap> Order(G); 4096 gap> Exponent(G); 4 gap> Rank(G); 2 gap> NilpotencyClassOfGroup(G); 5