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Problem 21 (Composing Permutations Using Disjoint Cycle Notation)

Let $R_{90}=(1,2,3,4)$ and $H=(1,2)(3,4)$.

  1. Express the compositions $R_{90}\circ H$ and $H\circ R_{90}$ in cycle notation. Be prepared to explain how you computed this.
  2. Express the compositions $R_{90}\circ (1,3)$ and $H\circ (1,3)$ in cycle notation. Can you see these products in the graph above?
  3. Express the composition $(1,2,4)\circ (1,4)(2,3)\circ (2,4)$ in cycle notation. Note that $(1,2,4)$ is not an automorphism of the square, but is a permutation.
  4. Evalute the Sage command below. In the second graph below you should see that starting at $()$ and following the colored arrows corresponding to $(2,4)$ and then $(1,4)(2,3)$ gets you to the element $(1,4,3,2)$ which is precisely the composition $(1,4)(2,3)\circ (2,4)$. In the third graph below, if you start at the identity $()$ and the follow the colored arrow corresponding to $ (2,4)$, and then $(1,4)(2,3)$, and then $(1,2,4)$, you should end up at the same spot as you did in part 3. Click evaluate again if the graph you have is hard to read (as the graph will redraw). 
g1 =  PermutationGroup(["(2,4)"])      #The set g1 is the span of the permutations listed.
d1=g1.cayley_graph()                   #This creates the Cayley graph of the g1. 
d1.show(color_by_label=True, vertex_size=0.03, vertex_labels=True) #This shows the Cayley graph. 
print(g1.list())                       #This print a list of the elements in g1. 

g2 = PermutationGroup(["(2,4)","(1,4)(2,3)"]) 
d2=g2.cayley_graph()
d2.show(color_by_label=True, vertex_size=0.03, vertex_labels=True)
print(g2.list())

g3 = PermutationGroup(["(2,4)","(1,4)(2,3)", "(1,2,4)"]) 
d3=g3.cayley_graph()
d3.show(color_by_label=True, vertex_size=0.03, vertex_labels=True,figsize=8)
print(g3.list())

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