# Burnside group:B(3,3)

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## Definition

This group is defined as the Burnside group $B(3,3)$. In other words, it is the quotient of free group:F3 under the relation that every element must cube to the identity.

## Arithmetic functions

Function Value Similar groups Explanation
underlying prime of p-group 3
order (number of elements, equivalently, cardinality or size of underlying set) 2187 groups with same order
exponent of a group 3 groups with same order and exponent of a group | groups with same exponent of a group
prime-base logarithm of order 7 groups with same prime-base logarithm of order
max-length of a group 7 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 7 chief length equals prime-base logarithm of order for group of prime power order
composition length 7 composition length equals prime-base logarithm of order for group of prime power order
prime-base logarithm of exponent 1 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 3 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

## 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 ]>```