<Chapter><Heading>Cohomology operations</Heading>
<Section><Heading>Steenrod operations on the classifying space of a finite <M>2</M>-group</Heading>

The following example determines a presentation for the cohomology ring
<M>H^\ast(Syl_2(M_{12}),\mathbb Z_2)</M>. The Lyndon-Hochschild-Serre spectral sequence, and Groebner basis routines from <B>Singular</B>, are used to determine how much of a resolution to compute for the presentation.

<Example>
<#Include SYSTEM "tutex/7.0.txt">
</Example>

The command <C>CohomologicalData(G,n)</C>
 prints complete information for the
cohomology ring <M>H^\ast(G, Z_2 )</M> of a <M>2</M>-group <M>G</M> 
provided that the integer <M>n</M> is
at least the maximal degree of a relator in a minimal set of relators for the
ring. Groebner basis routines from <B>Singular</B> are called involved in the example. 

<P/>The following example produces complete information on the Steenrod algebra of group number <M>8</M> in <B>GAP</B>'s library of groups of order <M>32</M>.


<Example>
<#Include SYSTEM "tutex/7.1.txt">
</Example>

</Section>
<Section><Heading>Steenrod operations on the classifying space of a finite <M>p</M>-group</Heading>

The following example constructs the first eight degrees of the
mod-<M>3</M> cohomology ring <M>H^\ast(G,\mathbb Z_3)</M>
for the group
<M>G</M> number 4 in <B>GAP</B>'s library of groups of order <M>81</M>.
It determines a minimal set of ring generators lying in degree <M>\le 8</M>
and it evaluates the Bockstein operator on these generators. 
Steenrod powers for <M>p\ge 3</M> are not implemented as  
no efficient method of implementation is known. 

<Example>
<#Include SYSTEM "tutex/7.2.txt">
</Example>

</Section>
</Chapter>
