Please Login to access more options.

Problem (Bonus) (The Point Group Of A Cube)

Consider a cube with vertices located at the 8 points $(\pm 1, 1,1)$, $(\pm 1, 1,-1)$, $(\pm 1, -1,1)$, $(\pm 1, -1,-1)$, and edges that connect each vertex to the three nearest points. For example, vertex $(1,1,1)$ has three edges, one connecting to each of $(1,1,-1)$, $(1,-1,1)$, and $(-1,1,1)$. We can view this as a graph whose vertices are these 8 points, along with the 12 edges. We will let $O_h$ represent the set of automorphisms of the graph of the cube.

  1. How many elements are in $O_h$? Explain.
  2. We will say that a set $S$ of permutations of $X$ is independent if for each $s\in S$, we have $\span (S\setminus\{s\})\neq \span (S)$. In other words, every element in $S$ is not in the span of the other permutations in $S$. Find an independent set $S$ of elements in $O_h$ so that $\span(S) = O_h$.
  3. Use sage to draw the Cayley graph of $O_h$ using S. In addition show the Cayley graph of $O_h$ using $S\setminus\{s\}$ for each $s\in S$. The code provided below can do this for you, by adding and/or removing elements from $S$.
  4. Challenge: Find another set $S_2$ with a different number of elements than $S$, so that $S_2$ is also an independent set whose span is $O_h$. 
S = ["(1,2,3,4)(5,6,7,8)","(1,5,8,4)(2,6,7,3)"]
g = PermutationGroup(S)    
d = g.cayley_graph()                                                 
print("There are " + str(order(g)) + " elements in the graph below.")
#To see a readable list of the elements, uncomment the line below.
#print(g.list()) 
d.show(color_by_label=True, vertex_size=0.03, vertex_labels=True)   

#We now make a graph of what happens if we remove one element.
for s in S:
    T=S[:]; #This makes a copy of S
    T.remove(s); #from which we remove s without affecting S. 
    gT=PermutationGroup(T)
    print("Removing " + s + " yields " + str(order(gT)) + " elements.")
    gT.cayley_graph().show(color_by_label=True, vertex_size=0.03, vertex_labels=True)

#The graph you create may be difficult to read in 2D. 
#Uncomment the next line to see a 3D Cayley graph. Note, it may cause your browser to hang for a moment. 
#d.show3d(color_by_label=True, vertex_size=0.03, vertex_labels=False)

For information on this group, see Wikipedia.

The following pages link to this page.