# Burnside group:B(2,4)

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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
Function Value Similar groups Explanation for function value
underlying prime of p-group 2
order (number of elements, equivalently, cardinality or size of underlying set) 4096 groups with same order
prime-base logarithm of order 12 groups with same prime-base logarithm of order
max-length of a group 12 max-length of a group equals prime-base logarithm of order for group of prime power order
chief length 12 chief length equals prime-base logarithm of order for group of prime power order
composition length 12 composition length equals prime-base logarithm of order for group of prime power order
exponent of a group 4 groups with same order and exponent of a group | groups with same exponent of a group By definition, it's the Burnside group of exponent 4.
prime-base logarithm of exponent 2 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 5 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 3 groups with same order and derived length | groups with same prime-base logarithm of order and derived length | groups with same derived length
Frattini length 3 groups with same order and Frattini length | groups with same prime-base logarithm of order and Frattini length | groups with same Frattini length
minimum size of generating set 2 groups with same order and minimum size of generating set | groups with same prime-base logarithm of order and minimum size of generating set | groups with same minimum size of generating set

## 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```