<Chapter><Heading>Group theoretic computations</Heading>
<Section><Heading>Third homotopy group of a supsension of an Eilenberg-MacLane space </Heading>

<P/>The following example uses the nonabelian tensor square of groups  to compute
the third homotopy group
<P/><M>\pi_3(S(K(G,1))) = \mathbb Z^{30}</M>
<P/>of the suspension of the Eigenberg-MacLane space <M>K(G,1)</M>
for <M>G</M> the free nilpotent group of class <M>2</M> on four generators.
<Example>
<#Include SYSTEM "tutex/5.1.txt">
</Example>

</Section>
<Section><Heading>Representations of knot quandles</Heading>

<P/> The following example constructs the finitely presented
quandles associated to the granny knot and square knot, and then computes
the number of quandle homomorphisms from these two finitely prresented quandles to the <M>17</M>-th quandle in <B>HAP</B>'s library of connected quandles of order <M>24</M>. The number of homomorphisms differs between the two cases. The computation therefore  establishes that the complement in <M>\mathbb R^3</M>
of the granny knot is not homeomorphic to the complement of the square knot.

<Example>
<#Include SYSTEM "tutex/5.2.txt">
</Example>

</Section>
<Section><Heading>Aspherical <M>2</M>-complexes</Heading>

<P/>The following example uses Polymake's linear programming routines to
establish  that the <M>2</M>-complex associated to the group presentation <M>&lt;x,y,z : xyx=yxy,\, yzy=zyz,\, xzx=zxz></M> is aspherical (that is, has contractible universal cover). The presentation is Tietze equivalent to the presentation used in the computer code, and the associated <M>2</M>-complexes are thus homotopy equivalent. 

<Example>
<#Include SYSTEM "tutex/5.3.txt">
</Example>


</Section>
<Section><Heading>Bogomolov multiplier</Heading>

<P/>The Bogomolov multiplier of a group is an isoclinism invariant. Using this property, the following example shows that there are precisely three groups of
order <M>243</M> with non-trivial Bogomolov multiplier. The groups in question are numbered 28, 29 and 30 in <B>GAP</B>'s library of small groups of order <M>243</M>.

<Example>
<#Include SYSTEM "tutex/5.4.txt">
</Example>

</Section>

</Chapter>
