# Burnside group:B(2,4)

From Groupprops

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## 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