# Burnside group:B(3,3)

## 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

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 2187#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 As Burnside group $B(d,3)$, d = 3: $3^{d + \binom{d}{2} + \binom{d}{3}}$ becomes $3^{3 + \binom{3}{2} + \binom{3}{3}} = 3^{3 + 3 + 1} = 3^7 = 2187$
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
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 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 exponent three implies class three for groups (further, the class is not two, hence it is exactly three)
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 follows from nilpotency class being three

## Elements

Further information: element structure of Burnside group:B(3,3)

### Summary

Item Value
order of the whole group (total number of elements) 2187, which is $3^7$
conjugacy class size statistics 1 (occurs 3 times), 3 (occurs 26 times), 27 (occurs 78 times)
maximum: 27, number of conjugacy classes: 107, lcm: 27
order statistics 1 of order 1, 2186 of order 3
maximum: 3, lcm (equals exponent of the whole group): 3

## Linear representation theory

Further information: linear representation theory of Burnside group:B(3,3)

### Summary

Item Value
degrees of irreducible representations over a splitting field (such as $\overline{\mathbb{Q}}$ or $\mathbb{C}$) 1 (occurs 27 times), 3 (occurs 78 times), 27 (occurs 2 times)
maximum: 27, lcm: 27, number: 107, sum of squares: 2187

## GAP implementation

### Group ID

This finite group has order 2187 and has ID 4487 among the groups of order 2187 in GAP's SmallGroup library. For context, there are groups of order 2187. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(2187,4487)

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

gap> G := SmallGroup(2187,4487);

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

IdGroup(G) = [2187,4487]

or just do:

IdGroup(G)

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

### 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 ]>