# Burnside group:B(3,3)

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## Definition

This group is defined as the Burnside group . In other words, it is the quotient of free group:F3 under the relation that every element must cube to the identity.

## Arithmetic functions

## GAP implementation

### Description by presentation

Below is a rather crude method. Probably, there is a more refined approach:

gap> F := FreeGroup(3);; gap> R1 := [Comm(F.2,Comm(F.2,F.1*F.3)),Comm(F.1 *F.3,Comm(F.2,F.1 * F.3))];; gap> R2 := [Comm(Comm(F.1,F.2),F.1),Comm(Comm(F.1,F.2),F.2)];; gap> R3 := [Comm(Comm(F.3,F.2),F.3),Comm(Comm(F.3,F.2),F.2)];; gap> R4 := [Comm(Comm(F.3,F.1),F.3),Comm(Comm(F.3,F.1),F.1)];; gap> R5 := [(F.1 * F.2)^3,(F.1 * F.3)^3, (F.2 * F.3)^3];; gap> R6 := [(F.1 * F.2^(-1))^3,(F.1 * F.3^(-1))^3, (F.2 * F.3^(-1))^3];; gap> R7 := [(F.1 * F.2 * F.3)^3];; gap> R8 := [(F.1 * F.2 * F.3^(-1))^3];; gap> R9 := [Comm(F.1,Comm(F.1,F.2*F.3)),Comm(F.2 *F.3,Comm(F.1,F.2 * F.3))];; gap> R10 := [Comm(F.2,Comm(F.2,F.1*F.3)),Comm(F.1 *F.3,Comm(F.2,F.1 * F.3))];; gap> R := Union(R1,R2,R3,R4,R5,R6,R7,R8,R9,R10); [ f1*f2^-1*f1*f2^-1*f1*f2^-1, f1*f2*f1*f2*f1*f2, f1*f3^-1*f1*f3^-1*f1*f3^-1, f1*f3*f1*f3*f1*f3, f2*f3^-1*f2*f3^-1*f2*f3^-1, f2*f3*f2*f3*f2*f3, f1^-1*f3^-1*f1*f3^-1*f1^-1*f3*f1*f3, f2^-1*f1^-1*f2*f1^-1*f2^-1*f1*f2*f1, f2^-1*f3^-1*f2*f3^-1*f2^-1*f3*f2*f3, f1*f2*f3^-1*f1*f2*f3^-1*f1*f2*f3^-1, f1*f2*f3*f1*f2*f3*f1*f2*f3, f1^-1*f3^-1*f1*f3*f1^-1*f3^-1*f1^-1*f3*f1^2, f2^-1*f1^-1*f2*f1*f2^-1*f1^-1*f2^-1*f1*f2^2, f2^-1*f3^-1*f2*f3*f2^-1*f3^-1*f2^ -1*f3*f2^2, f1^-1*f3^-1*f2^-1*f1^-1*f2*f3*f1*f3^-1*f2^-1*f1*f2*f3, f2^-1*f3^-1*f1^-1*f2^-1*f1*f3*f2*f3^-1*f1^-1*f2*f1*f3, f3^-1*f1^-1*f3^-1*f1^-1*f2^-1*f1*f3*f2*f1*f3*f2^-1*f3^-1*f1^-1*f2*f1*f3, f3^-1*f2^-1*f3^-1*f2^-1*f1^-1*f2*f3*f1*f2*f3*f1^-1*f3^-1*f2^-1*f1*f2*f3 ] gap> G := F/R; <fp group on the generators [ f1, f2, f3 ]>