Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan 20 Cayley’s Theorem We have already met (i.e. Section 6) the symmetric group Sym(S), the group of all permutations on a set S. It was one of our first examples of a group. In fact it is a very important group, partly because of Cayley’s theorem which we discuss in this section. Cayley’s theorem represents a group as a subgroup of a permutation group(up to an isomorphism). This is often advantageous, because permutation groups are fairly concrete objects. For example, it’s straightfroward to write pro- grams to do arithmetic in finite permutation groups. Theorem 20.1 (Cayley’s Theorem) Any group G is isomorphic to a subgroup of Sym(G). Proof. Given g ∈ G, we define a map λg : G → G by λg(x) = gx for all x ∈ G. This is a well-defined mapping. Indeed, if x = y then gx = gy so that λg(x) = λg(y). Next, we show that λg is one-to-one. To see this, suppose that λg(x) = λg(y). Then gx = gy and by the left-cancellation property −1 −1 x = y. To see that λg is onto, let y ∈ G. Then g y ∈ G and λg(g y) = y. Hence, λg ∈ Sym(G). Next, We define Λ : G → Sym(G) by Λ(g) = λg. This is a well-defined mapping. For if g1 = g2 then g1x = g2x for all x ∈ G, that is, λg1 (x) = λg2 (x) for all x ∈ G and hence λg1 = λg2 , i.e. Λ(g1) = Λ(g2). Now, given g1, g2 ∈ G we have λg1g2 (x) = (g1g2)x = g1(g2x) = λg1 (g2x) = λg1 λg2 (x) for all x ∈ G. Thus, Λ(g1g2) = λg1g2 = λg1 λg2 = Λ(g1)Λ(g2), and so Λ is a homomorphism. Finally, we show that Λ is one-to-one. Indeed, if Λ(g1) = Λ(g2) then λg1 (x) = λg2 (x) for all x ∈ G. In particular, λg1 (e) = λg2 (e). That is, g1e = g2e or g1 = g2. By Theorem 18.2(v), G ≈ Λ(G). Corollary 20.1 Every finite group G of order n is isomorphic to a subgroup of Sn. 1 Proof. Listing the element of G as G = {a1, a2, ··· , an}. The proof of Theo- a1 ··· an rem 20.1 assigns to the element ai the map λai = . Now, aia1 ··· aian aia1, ··· , aian is just the ith row of the Cayley table for G, and thus is sim- ply a rearrangement of a1, ··· , an, say aθ (1), ··· , aθ (n) where θi ∈ Sn; so i i a1 ··· an λai = . If we now replace ai by i, we map λai to aθi(1) ··· aθi(n) 1 ··· n = θi. By Theorem 20.1, {θ1, θ2, ··· , θn} ≈ Λ(G) ≈ G. θi(1) ··· θi(n) That is, G is isomorphic to a subgroup of Sn. Example 20.1 Consider the following Cayley table of a group G = {e, a, b, c}. V e a b c e e a b c a a e c b b b c e a c c b a e We then have e a b c 1 2 3 4 λ = , θ = = (1) e e a b c 1 1 2 3 4 e a b c 1 2 3 4 λ = , θ = = (12)(34) a a e c b 2 2 1 4 3 e a b c 1 2 3 4 λ = , θ = = (13)(24) b b c e a 3 3 4 1 2 e a b c 1 2 3 4 λ = , θ = = (14)(23) c c b a e 4 4 3 2 1 Hence G is isomorphic to the subgroup {(1), (1 2)(3 4), (1 3)(2 4), (1 4)(2 3)} = h(1 2)(3 4), (1 3)(2 4)i of S4. 2.