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9. Commutator and nonabelian tensor computations

### 9. Commutator and nonabelian tensor computations

`BaerInvariant(G,c)`

Inputs a nilpotent group G and integer c>0. It returns the Baer invariant M^(c)(G) defined as follows. For an arbitrary group G let L^*_c+1(G) be the (c+1)-st term of the upper central series of the group U=F/[[[R,F],F]...] (with c copies of F in the denominator) where F/R is any free presentation of G. This is an invariant of G and we define M^(c)(G) to be the kernel of the canonical homomorphism M^(c)(G) --> G. For c=1 the Baer invariant M^(1)(G) is isomorphic to the second integral homology H_2(G,Z).

This function requires the NQ package.

`Coclass(G)` Inputs a group G of prime-power order p^n and nilpotency class c say. It returns the integer r=n-c .

`EpiCentre(G,N)`
`EpiCentre(G)`

Inputs a finite group G and normal subgroup N and returns a subgroup Z^*(G,N) of the centre of N. The group Z^*(G,N) is trivial if and only if there is a crossed module d:E --> G with N=Image(d) and with Ker(d) equal to the subgroup of E consisting of those elements on which G acts trivially.

If no value for N is entered then it is assumed that N=G. In this case the group Z^*(G,G) is trivial if and only if G is isomorphic to a quotient G=E/Z(E) of some group E by the centre of E. (See also the command UpperEpicentralSeries(G,c). )

`NonabelianExteriorProduct(G,N)`

Inputs a finite group G and normal subgroup N. It returns a record E with the following components.

- E.homomorphism is a group homomorphism µ : (G wedge N) --> G from the nonabelian exterior product (G wedge N) to G. The kernel of µ is the relative Schur multiplier.

- E.pairing(x,y) is a function which inputs an element x in G and an element y in N and returns (x wedge y) in the exterior product (G wedge N) .

This function should work for reasonably small nilpotent groups or extremely small non-nilpotent groups.

`NonabelianSymmetricKernel(G)`
`NonabelianSymmetricKernel(G,m)`

Inputs a finite or nilpotent infinite group G and returns the abelian invariants of the Fourth homotopy group SG of the double suspension SSK(G,1) of the Eilenberg-Mac Lane space K(G,1).

For non-nilpotent groups the implementation of the function NonabelianSymmetricKernel(G) is far from optimal and will soon be improved. As a temporary solution to this problem, an optional second variable m can be set equal to 0, and then the function efficiently returns the abelian invariants of groups A and B such that there is an exact sequence 0 --> B --> SG --> A --> 0.

Alternatively, the optional second varible m can be set equal to a positive multiple of the order of the symmetric square (G tildeotimes G). In this case the function returns the abelian invariants of SG. This might help when G is solvable but not nilpotent (especially if the estimated upper bound m is reasonable accurate).

`NonabelianSymmetricSquare(G)`
`NonabelianSymmetricSquare(G,m)`

Inputs a finite or nilpotent infinite group G and returns a record T with the following components.

- T.homomorphism is a group homomorphism µ : (G tildeotimes G) --> G from the nonabelian symmetric square of G to G. The kernel of µ is isomorphic to the fourth homotopy group of the double suspension SSK(G,1) of an Eilenberg-Mac Lane space.

- T.pairing(x,y) is a function which inputs two elements x, y in G and returns the tensor (x otimes y) in the symmetric square (G otimes G) .

An optional second varible m can be set equal to a multiple of the order of the symmetric square (G tildeotimes G). This might help when G is solvable but not nilpotent (especially if the estimated upper bound m is reasonable accurate) as the bound is used in the solvable quotient algorithm.

The optional second variable m can also be set equal to 0. In this case the Todd-Coxeter procedure will be used to enumerate the symmetric square even when G is solvable.

This function should work for reasonably small solvable groups or extremely small non-solvable groups.

`NonabelianTensorProduct(G,N)`

Inputs a finite group G and normal subgroup N. It returns a record E with the following components.

- E.homomorphism is a group homomorphism µ : (G otimes N ) --> G from the nonabelian exterior product (G otimes N) to G.

- E.pairing(x,y) is a function which inputs an element x in G and an element y in N and returns (x otimes y) in the tensor product (G otimes N) .

This function should work for reasonably small nilpotent groups or extremely small non-nilpotent groups.

`NonabelianTensorSquare(G)`
`NonabelianTensorSquare(G,m)`

Inputs a finite or nilpotent infinite group G and returns a record T with the following components.

- T.homomorphism is a group homomorphism µ : (G otimes G) --> G from the nonabelian tensor square of G to G. The kernel of µ is isomorphic to the third homotopy group of the suspension SK(G,1) of an Eilenberg-Mac Lane space.

- T.pairing(x,y) is a function which inputs two elements x, y in G and returns the tensor (x otimes y) in the tensor square (G otimes G) .

An optional second varible m can be set equal to a multiple of the order of the tensor square (G otimes G). This might help when G is solvable but not nilpotent (especially if the estimated upper bound m is reasonable accurate) as the bound is used in the solvable quotient algorithm.

The optional second variable m can also be set equal to 0. In this case the Todd-Coxeter procedure will be used to enumerate the tensor square even when G is solvable.

This function should work for reasonably small solvable groups or extremely small non-solvable groups.

`RelativeSchurMultiplier(G,N)` Inputs a finite group G and normal subgroup N. It returns the homology group H_2(G,N,Z) that fits into the exact sequence ...--> H_3(G,Z) --> H_3(G/N,Z) --> H_2(G,N,Z) --> H_3(G,Z) --> H_3(G/N,Z) --> .... This function should work for reasonably small nilpotent groups G or extremely small non-nilpotent groups.
`TensorCentre(G)` Inputs a group G and returns the largest central subgroup N such that the induced homomorphism of nonabelian tensor squares (G otimes G) --> (G/N otimes G/N) is an isomorphism. Equivalently, N is the largest central subgroup such that pi_3(SK(G,1)) --> pi_3(SK(G/N,1)) is injective.

`ThirdHomotopyGroupOfSuspensionB(G)`
`ThirdHomotopyGroupOfSuspensionB(G,m)`

Inputs a finite or nilpotent infinite group G and returns the abelian invariants of the third homotopy group JG of the suspension SK(G,1) of the Eilenberg-Mac Lane space K(G,1).

For non-nilpotent groups the implementation of the function ThirdHomotopyGroupOfSuspensionB(G) is far from optimal and will soon be improved. As a temporary solution to this problem, an optional second variable m can be set equal to 0, and then the function efficiently returns the abelian invariants of groups A and B such that there is an exact sequence 0 --> B --> JG --> A --> 0.

Alternatively, the optional second varible m can be set equal to a positive multiple of the order of the tensor square (G otimes G). In this case the function returns the abelian invariants of JG. This might help when G is solvable but not nilpotent (especially if the estimated upper bound m is reasonable accurate).

`UpperEpicentralSeries(G,c)`

Inputs a nilpotent group G and an integer c. It returns the c-th term of the upper epicentral series 1 < Z_1^*(G) < Z_2^*(G) < ....

The upper epicentral series is defined for an arbitrary group G. The group Z_c^* (G) is the image in G of the c-th term Z_c(U) of the upper central series of the group U=F/[[[R,F],F] ... ] (with c copies of F in the denominator) where F/R is any free presentation of G.

This functions requires the NQ package.

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