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.

Solution

Here are some fun sets to use for $S$.

  • S = ["(1,3)(2,4)(5,7)(6,8)","(2,5,4)(6,8,3)","(1,6,3)(5,7,4)"] (Two rotations about a diagaonal, and 180 degree rotation about an axis.
  • S = ["(1,6)(4,7)","(1,3)(2,4)(5,7)(6,8)","(2,5,4)(6,8,3)"] (Reflection about plane through middle, a 180 degree rotation, and a rotation about a diagonal)
  • S = ["(1,2)","(2,3)","(3,4)","(5,6)"] (think of this as an octohedron, with 1-4 in the middle, and 5,6 at the top and bottom.
  • S = ["(1,5)(2,6)(3,7)(4,8)","(1,2)(3,4)(5,6)(7,8)","(1,4)(2,3)(5,8)(6,7)"]
  • S = ["(1,5)(2,6)(3,7)(4,8)","(1,2)(3,4)(5,6)(7,8)","(1,4)(2,3)(5,8)(6,7)","(1,7)(2,8)(3,5)(4,6)"]
  • S = ["(1,5)(2,6)(3,7)(4,8)","(1,2)(3,4)(5,6)(7,8)","(1,4)(2,3)(5,8)(6,7)","(1,7)"]
  • S = ["(1,5)(2,6)(3,7)(4,8)","(1,2)(3,4)(5,6)(7,8)","(1,4)(2,3)(5,8)(6,7)","(1,7)(2,3)(6,4)(5,8)"]
  • S = ["(1,2)(3,4)(5,6)(7,8)","(1,4)(2,3)(5,8)(6,7)","(1,7)(2,3)(6,4)(5,8)","(2,5,4)(6,8,3)"]
Tags
  • When you are ready to submit this written work for grading, add the phrase [[!Submit]] to your page. This will tell me that you have completed the page (it's past rough draft form, and you believe it is in final draft form). Don't type [[!Submit]] on a rough draft.
  • If I put [[!NeedsWork]] on your page, then your job is to review what I've written, address any comments made, and then delete all the comments I made. When you have finished reviewing your work, leave [[!NeedsWork]] on your page and type [[!Submit]]. (Both tags will show up). This tells me you have addressed the comments.
  • I'll mark your work with [[!Complete]] after you have made appropriate revisions.