Project #5755 - Automorphisms

 

Consider a 4-cycle, G, with vertices V = {A; B; C; D}, and edges {A; B},

{B; C}, {C; D}, {D; A}. Note that you can have an isomorphism that maps G

to itself. Such a map is called an automorphism.

The most obvious automorphism of G maps A to A, B to B, C to C, and

D to D. This is called the identity map. But there are less trivial ones: 

 

for example, let f map A to B, B to C, C to D, and D to A.

 

Extra Credit Exercise 1: Find all 8 automorphims of G.

Note that if you have two automorphisms, f and g, then f  g (which just means ?rst do g then do f) is also an automorphism. For example, if f is the automorphism given above, (f  f)(A) = f(f(A)) = f(B) = C, (f  f)(B) =

f(f(B)) = f(C) = D, and so on. The automorphisms of G form a group

with this operation. Recall a group operation has an identity, (in this case, the

identity map given above), and inverses.

Extra Credit Exercise 2: What is the inverse of f? That is, of the eight

automorphisms you listed in the previous exercise which of them is a map g

such that f  g is the identity map.

Extra Credit Exercise 3: Find two automorphisms of G, say g1 and g2 so

that g1  g2 6= g2  g1.

Extra Credit Exercise 4: Find two automorphism which generate all eight

automorphisms of G. Draw a Cayley graph of the automorphisms of G.

Extra Credit Exercise 5: Give a proof by induction that an in-order traversal of a binary search tree puts the things stored in the search tree in order

Subject Mathematics
Due By (Pacific Time) 05/07/2013 10:00 am
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