1 Cayley Theorem

1. Lets define a group

• \(G = { a b c d e}\)
  

2. Pick an element of that group let’s say \(c\).
   Left multiply \(G\) by \(c\)

• \(cG = { ca cb cc cd ce}\)
• \(cG = { b e a c d}\)

3. Notice the resulting multiplication just created a permutation of G.

• \(cG \in Perm(G)\)
  • \(Perm(G)\) = Set containing different permutations of G including group G itself

4. We can define the left multiplication of \(G\) as a higher order function f

• f :: G -> {G} -> Perm(G)
• f(c) = cG for our specific example
• function f takes a group element (in our example \(c\)) and our Group G and returns a permutation of G.

5. We can curry f to lambda x, f(x) :: {G} -> Perm(G)

• In abstract algebra this means that Group $G4 is isomorphic to a subgroup \(S_G\) which is termed the Permutation group.