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# 5 Schur extensions for p-power-poly-pcp-groups

### Sections

In this chapter we describe how the consistent pp-presentations of infinite coclass sequences can be used to compute a pp-presentation for the corresponding Schur extensions (see EF11).

For a group G = F/R the Schur extension H is defined as H = F/[F,R] (see EN08).

So for a parameter x that can take values in the positive integers, let (Gx = F/Rx | xN), for N the positive integers, describe an infinite coclass sequence of finite p-groups GX of coclass r. Then for each value for the parameter x, the group Gx has a consistent polycyclic presentation with generators g1, ·.·, gn, t1, ·.·, td and relations

 gip = rel[i][i],
 tiexpo = rel[n+i][n+i],
 gigj = rel[j][i],
 tigj = rel[j][n+i],
 titj = 1·

Then we compute a consistent pp-presentation of the corresponding Schur extensions of with generators g1, ·.·, gn, t1, ·.·, td, c1, ·.·cm and relations

 gip=rel[i][i],
 tiexpo = rel[n+i][n+i],
 ciexpo_vec[i] = rel[n+d+i,n+d+i],
 gigj = rel[j][i],
 tigj = rel[j][n+i],
 titj = rel[n+j][n+i],
 cigj = 1,
 citj = 1,
 cicj = 1·

where the ti's commute modulo c1, ·.·, cm and the ci's are central.

## 5.1 Computing Schur extensions

• `SchurExtParPres( `G` )`

computes the Schur extensions corresponding to the p-power-poly-pcp-groups G and returns them as p-power-poly-pcp-groups.

• `SchurExtParPres( `ParPres` ) F`

computes a consistent pp-presentation of Schur extensions of the groups defined by the record ParPres which describes p-power-poly-pcp-groups. The output is a record rec(rel, expo, n, d, m, prime, cc, expo_vec, name), which describes the Schur extensions as p-power-poly-pcp-groups; it is encoded in a form that it can be used as input for PPPPcpGroups.

```gap> SchurExtParPres( ParPresGlobalVar_2_1 );
rec( prime := 2,
rel := [ [ [ [ 7, 1 ] ] ], [ [ [ 2, 1 ], [ 3, -1+2*2^x ], [ 6, 1-2*2^x ] ],
[ [ 3, 1 ], [ 5, 1 ] ] ],
[ [ [ 3, -1+2*2^x ], [ 4, 1 ], [ 6, 2-2*2^x ] ], [ [ 3, 1 ] ],
[ [ 4, 1 ], [ 6, 2*2^x ] ] ],
[ [ [ 4, 1 ] ], [ [ 4, 1 ] ], [ [ 4, 1 ] ], [ [ 4, 0 ] ] ],
[ [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 1 ] ], [ [ 5, 0 ] ] ]
,
[ [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ], [ [ 6, 1 ] ],
[ [ 6, 0 ] ] ],
[ [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ], [ [ 7, 1 ] ],
[ [ 7, 1 ] ], [ [ 7, 0 ] ] ] ], n := 2, d := 1, m := 4,
expo := 2*2^x, expo_vec := [ 2, 0, 0, 0 ], cc := fail, name := "SchurExt_D"
)
```

## 5.2 Computing other invariants from Schur extensions

• `AbelianInvariantsMultiplier( `G` ) F`

computes the abelian invariants of the Schur multiplicators M(G) of the p-power-poly-pcp-groups G. The output is a list [d1, ·.·, dk] consisting elements di, depending on the underlying parameter, such that M(G) ≅ Cd1 ×…×Cdk.

```gap> G := PPPPcpGroups( ParPresGlobalVar_2_1 );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> AbelianInvariantsMultiplier( G );
[ 2 ]
```

• `SchurMultiplicatorPPPPcps( `G` ) F`

computes the Schur multiplicators of the p-power-poly-pcp-groups G and then returns the corresponding PPPPcpGroups.

```gap> G := PPPPcpGroups( ParPresGlobalVar_3_1 );
< P-Power-Poly pcp-group with 5 generators of relative orders [ 3,3,3,3*3^x,3*3^x ] >
gap> SchurMultiplicatorPPPPcps( G );
< P-Power-Poly-pcp-groups with 2 generators of relative orders [ 3,9*3^x ] >
```

• `AbelianInvariants( `G` ) F`

computes the abelian invariants of the p-power-poly-pcp-groups G and returns them as a list of list describing the parametrised elements.

```gap> G := PPPPcpGroups( ParPresGlobalVar_2_1 );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> AbelianInvariants( G );
[ 2, 2 ]
```

• `ZeroCohomologyPPPPcps( `G`[, `p`] ) F`

computes the zero-th-cohomology groups H0(G,R) of the p-power-poly-pcp-groups G with coefficients in R, where RGF(p) if the prime p is given or RZ otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a1,…, ak] where the cohomology group is isomorphic to Ca1 ×…×Cak with Ci a cyclic group of order i (for i > 0) and C0 is interpreted as Z.

```gap> G := PPPPcpGroups( ParPresGlobalVar_2_1 );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> ZeroCohomologyPPPPcp( G, 2 );
[ 2 ]
```

• `FirstCohomologyPPPPcps( `G`[, `p`] ) F`

computes the first-cohomology groups H1(G,R) of the p-power-poly-pcp-groups G with coefficients in R, where RGF(p) if the prime p is given or RZ otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a1,…, ak] where the cohomology group is isomorphic to Ca1 ×…×Cak with Ci a cyclic group of order i (for i > 0) and C0 is interpreted as Z.

```gap> G := PPPPcpGroups( ParPresGlobalVar_2_1 );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> FirstCohomologyPPPPcps( G );
[  ]
```

• `SecondCohomologyPPPPcps( `G`[, `p`] ) F`

computes the second-cohomology groups H2(G,R) of the p-power-poly-pcp-groups G with coefficients in R, where RGF(p) if the prime p is given or RZ otherwise. The action of G on R is taken to be trivial. The function returns a list of integers [a1,…, ak] where the cohomology group is isomorphic to Ca1 ×…×Cak with Ci a cyclic group of order i (for i > 0) and C0 is interpreted as Z.

```gap> G := PPPPcpGroups( ParPresGlobalVar_2_1 );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> SecondCohomologyPPPPcps( G, 2 );
[ 2, 2, 2 ]
```

## 5.3 Info classes for the computation of the Schur extension

The following info classes are available

• `InfoConsistencyRelPPowerPoly V`

`level 1`
shows which consistency relations are computed and gives the result;

the default value is 0.

• `InfoCollectingPPowerPoly V`

`level 1`
shows what is done during collecting;

the default value is 0.

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SymbCompCC manual
March 2018