Cayley’s Theorem Math 281 Theorem Every finite group exists as a subgroup of some symmetric group Sd . In other words, if G is a finite group with d elements, there exists an injective homomorphism ϕ : G −→ Sd One of the beautiful features of this theorem is that the prove is constructive. Given any group, you can actually naturally define such an injective homomorphism. This worksheet is meant to accompany the formal proof of the theorem with a couple of exercises that illustrate the spirit of the general proof. 1 Case 1: Z/4Z Consider the group Z/4Z. In the figure below, draw the appropriate arrows between the elements of the bubbles. The way you should draw the arrows is as follows. On the upper left corner of each pair of bubbles you see the “instructions: +0 means you should connect each element appearing in the left bubble to itself +0 on the right bubble, etc etc. Then notice that you have a permutation of the set {1, 2, 3, 4 = 0}. Write this permutation in cycle notation on the line to the right. 2 Case 2: Klein group Consider the group G = {1, a, b, c}. In the figure below, draw the appropriate arrows between the elements of the bubbles. The way you should draw the arrows is as follows. On the upper left corner of each pair of bubbles you see the “instructions: a means you should connect each element appearing in the left bubble to itself multiplied by a on the right bubble, etc etc. Then notice that you have a permutation of the set {1, a, b, c}. Relabel these elements however you want with the numbers 1, 2, 3, 4 and write this permutation in cycle notation on the line to the right. 1 +0 1 2 3 4=0 1 2 3 0=4 +1 1 2 3 4=0 1 2 3 0=4 +2 1 2 3 4=0 1 2 3 0=4 +3 1 2 3 4=0 1 2 3 0=4 2 1 1 1 a a b b c c a 1 1 a a b b c c b 1 a b c 1 a b c c 1 a b c 1 a b c 3