====== Soccerballahedron ======
==== Problem ====
Construct the [[definition:Cayley graph]] of the group defined by the [[definition:group presentation]] $$\left< a,b\mid a^2, b^5, ababab\right>.$$ 
Can you think of a common object whose shape is similar to this graph?

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==== Remarks ====  
  * One option ends up with a soccerball.
  * This is a presentation for the [[definition:alternating group]] $A_5$. 
  * I decided to omit $=e$ from each relator. 
<sagecell>
g = AlternatingGroup(5)
d = g.cayley_graph(); d
d.show(color_by_label=True, vertex_size=0.03, vertex_labels=False)
show(g.semigroup_generators())
show(g.objgens())
#d.show3d(color_by_label=True, edge_size=0.01, edge_size2=0.02, vertex_size=0.03)
#Uncomment the last line if you would like to allow java to draw the graph in 3D. 
#
</sagecell>

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==== $\LaTeX$ version ====
<code>
%%%%%
% DEPENDENCIES
% RequiredPackages: \usepackage{tikz}
% RequiredMacros: \DeclareMathOperator{\aut}{Aut} 
%%%%%
\begin{problem}
Type the problem code here.
\end{problem}
</code>

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==== External links ====
  * [[http://web.bentley.edu/empl/c/ncarter/vgt/gallery-4.html]]
  * [[http://www.lbl.gov/Science-Articles/Archive/MSD-C60-molecular-layer.html]]

{{tag>problem ben needsreview rben}}