%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% %W schurextension.tex GAP documentation Dörte Feichtenschlager %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Chapter{Schur extensions for p-power-poly-pcp-groups}
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 \cite{EF11}).
For a group $G = F/R$ the Schur extension $H$ is defined as $H = F/[F,R]$
(see \cite{EN08}).
So for a parameter <x> that can take values in the positive integers, let
$(G_x = F/R_x | x \in\N)$, for $\N$ the positive integers, describe an
infinite coclass sequence of finite $p$-groups $G_X$ of coclass $r$. Then for
each value for the parameter <x>, the group $G_x$ has a consistent polycyclic
presentation with generators $g_1, ..., g_n, t_1, ..., t_d$ and relations
Then we compute a consistent pp-presentation of the corresponding Schur
extensions of with generators $g_1, ..., g_n, t_1, ..., t_d, c_1, ... c_m$ and
relations
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 %display{tex}
{\tt PPPPcpGroups}, %enddisplay "PPPPcpGroups".
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{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 $[d_1, ..., d_k]$
consisting elements $d_i$, depending on the underlying parameter, such that
$M(G) \cong C_{d_1} \times\ldots\times C_{d_k}$.
\beginexample
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> AbelianInvariantsMultiplier( G );
[ 2 ] \endexample
\>SchurMultiplicatorPPPPcps( <G> )!{for p-power-poly-pcp-groups} F
computes the Schur multiplicators of the <p>-power-poly-pcp-groups <G> and
then returns the corresponding %display{tex}
{\tt PPPPcpGroups}, %enddisplay "PPPPcpGroups".
\beginexample
gap> G := PPPPcpGroups( ParPresGlobalVar_3_1[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 ] > \endexample
\>AbelianInvariants( <G> )!{for p-power-poly-pcp-groups} 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.
\beginexample
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> AbelianInvariants( G );
[ 2, 2 ] \endexample
\>ZeroCohomologyPPPPcps( <G>[, <p>] ) F
computes the zero-th-cohomology groups $H^0(G,R)$ of the
<p>-power-poly-pcp-groups <G> with coefficients in $R$, where $R \cong GF(p)$ if
the prime $p$ is given or $R \cong\Z$ otherwise. The action of $G$ on $R$ is
taken to be trivial. The function returns a list of integers $[a_1,\ldots,
a_k]$ where the cohomology group is isomorphic to $C_{a_1} \times\ldots \times C_{a_k}$ with $C_i$ a cyclic group of order $i$ (for $i > 0$) and $C_0$
is interpreted as $\Z$.
\beginexample
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> ZeroCohomologyPPPPcp( G, 2 );
[ 2 ] \endexample
\>FirstCohomologyPPPPcps( <G>[, <p>] ) F
computes the first-cohomology groups $H^1(G,R)$ of the
<p>-power-poly-pcp-groups <G> with coefficients in $R$, where $R \cong GF(p)$ if
the prime $p$ is given or $R \cong\Z$ otherwise. The action of $G$ on $R$ is
taken to be trivial. The function returns a list of integers $[a_1,\ldots,
a_k]$ where the cohomology group is isomorphic to $C_{a_1} \times\ldots \times C_{a_k}$ with $C_i$ a cyclic group of order $i$ (for $i > 0$) and $C_0$
is interpreted as $\Z$.
\beginexample
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> FirstCohomologyPPPPcps( G );
[ ] \endexample
\>SecondCohomologyPPPPcps( <G>[, <p>] ) F
computes the second-cohomology groups $H^2(G,R)$ of the
<p>-power-poly-pcp-groups <G> with coefficients in $R$, where $R \cong GF(p)$ if
the prime $p$ is given or $R \cong\Z$ otherwise. The action of $G$ on $R$ is
taken to be trivial. The function returns a list of integers $[a_1,\ldots,
a_k]$ where the cohomology group is isomorphic to $C_{a_1} \times\ldots \times C_{a_k}$ with $C_i$ a cyclic group of order $i$ (for $i > 0$) and $C_0$
is interpreted as $\Z$.
\beginexample
gap> G := PPPPcpGroups( ParPresGlobalVar_2_1[1] );
< P-Power-Poly-pcp-groups with 3 generators of relative orders [ 2,2,2*2^x ] >
gap> SecondCohomologyPPPPcps( G, 2 );
[ 2, 2, 2 ] \endexample
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Info classes for the computation of the Schur extension}
The following info classes are available
\>`InfoConsistencyRelPPowerPoly' V
\beginitems
`level 1' & shows which consistency relations are computed and gives the
result; \enditems
the default value is 0.
\>`InfoCollectingPPowerPoly' V
\beginitems
`level 1' & shows what is done during collecting; \enditems
the default value is 0.
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