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11 Commutator and nonabelian tensor computations

### 11 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.

`BogomolovMultiplier(G)`
`BogomolovMultiplier(G, "standard")` `BogomolovMultiplier(G, "homology")` `BogomolovMultiplier(G, "tensor")`

Inputs a finite group G and returns the quotient H_2(G,Z)/K(G) of the second integral homology of G where K(G) is the subgroup of H_2(G,Z) generated by the images of all homomorphisms H_2(A,Z) → H_2(G,Z) induced from abelian subgroups of G.

Three slight variants of the implementation are available. The defaults "standard" implementation seems to work best on average. But for some groups the "homology" implementation or the "tensor" implementation will be faster. The variants are called by including the appropriate string as the second argument.

`Bogomology(G,n)` Inputs a finite group G and positive integer n, and returns the quotient H_n(G,Z)/K(G) of the degree n integral homology of G where K(G) is the subgroup of H_n(G,Z) generated by the images of all homomorphisms H_n(A,Z) → H_n(G,Z) induced from abelian subgroups of G.
`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 ∧ N) ⟶ G from the nonabelian exterior product (G ∧ 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 ∧ y) in the exterior product (G ∧ 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 tilde⊗ 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 tilde⊗ 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 ⊗ y) in the symmetric square (G ⊗ G) .

An optional second varible m can be set equal to a multiple of the order of the symmetric square (G tilde⊗ 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 ⊗ N ) ⟶ G from the nonabelian exterior product (G ⊗ 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 ⊗ y) in the tensor product (G ⊗ 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 ⊗ 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 ⊗ y) in the tensor square (G ⊗ G) .

An optional second varible m can be set equal to a multiple of the order of the tensor square (G ⊗ 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 ⊗ G) ⟶ (G/N ⊗ G/N) is an isomorphism. Equivalently, N is the largest central subgroup such that π_3(SK(G,1)) ⟶ π_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 ⊗ 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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