Math 3230 Abstract Algebra I Sec 5.3: Examples of group actions Slides created by M. Macauley, Clemson (Modified by E. Gunawan, UConn) http://egunawan.github.io/algebra Abstract Algebra I Sec 5.3 Examples of group actions Abstract Algebra I 1 / 10 Groups acting on elements, subgroups, and cosets It is often of interest to analyze the action of a group G on its elements, subgroups, or cosets of some fixed H ≤ G. Sometimes, the orbits and stabilizers of these actions are algebraic objects you have seen before. Sometimes a deep theorem has a slick proof via a clever group action. For example, we will see how Cayley's theorem (every group G is isomorphic to a group of permutations) follows immediately once we look at the correct action. Here are common examples of group actions: a. G acts on itself by right-multiplication (or left-multiplication). b. G acts on itself by conjugation. c. G acts on its subgroups by conjugation (not covered). d. G acts on the right-cosets of a fixed subgroup H ≤ G by right-multiplication. For each of these, we'll analyze the orbits, stabilizers, and fixed points. Sec 5.3 Examples of group actions Abstract Algebra I 2 / 10 Recall Example 3: Groups acting on themselves by right-multiplication We've seen how groups act on themselves by right-multiplication (Example 3 in Slides 5.1). While this action is boring (any Cayley diagram is an action diagram!), it leads to a slick proof of Cayley's theorem. Theorem 2 (Cayley's theorem) If jGj = n, then there is an injective homomorphism G ,! Sn. Proof. The group G acts on itself (that is, S = G) by right-multiplication: φ: G −! Perm(S) =∼ Sn ; φ(g) is the permutation that sends each x 7! xg. There is only one orbit: G = S. The stabilizer of any x 2 G is just the identity element: Stab(x) = fg 2 G j xg = xg = feg : \ Therefore, the kernel of this action is Ker φ = Stab(x) = feg. x2G Since Ker φ = feg, the homomorphism φ is injective. Sec 5.3 Examples of group actions Abstract Algebra I 3 / 10 Example 5: Groups acting on themselves by conjugation Another way a group G can act on itself (that is, S = G) is by conjugation: φ: G −! Perm(S) ; φ(g) is the permutation that sends each x 7! g −1xg. The orbit of x 2 G is its conjugacy class: −1 Orb(x) = fx.φ(g) j g 2 Gg = fg xg j g 2 Gg = clG (x) : The stabilizer of x is the set of elements that commute with x; called its centralizer: −1 Stab(x) = fg 2 G j g xg = xg = fg 2 G j xg = gxg := CG (x) The fixed points of φ are precisely those in the center of G: Fix(φ) = fx 2 G j g −1xg = x for all g 2 Gg = Z(G) : By the Orbit-Stabilizer theorem, jGj = j Orb(x)j · j Stab(x)j = j clG (x)j · jCG (x)j. Thus, we immediately get the following new result about conjugacy classes: Theorem 3 For any x 2 G, the size of the conjugacy class clG (x) divides the size of G. Sec 5.3 Examples of group actions Abstract Algebra I 4 / 10 Example 5a: S3 acting on itself by conjugation Consider the action of G = S3 on itself by conjugation. The orbits of the action are the conjugacy classes: feg, f permutations of the form (ab)g, f permutations of the form (abc)g The fixed points of φ are elements of the size-1 conjugacy classes, only e has a size-one conjugacy classs. By the Orbit-Stabilizer theorem: jS j 6 j Stab(x)j = 3 = : j Orb(x)j j clG (x)j The stabilizer subgroups are as follows: Stab(e) = S3, Stab( (123) ) = Stab( (132) ) = h(123)i =∼ C3, Stab( (12) ) = fe; (12)g =∼ C2 Stab( (13) ) = Stab( (23) ) = Sec 5.3 Examples of group actions Abstract Algebra I 5 / 10 Example 5b: D6 acting on itself by conjugation Consider the action of G = D6 on itself by conjugation. e r 2 2 4 The orbits of the action are the r f r f r f conjugacy classes: r 3 r 5 r 4 rf r 3f r 5f The fixed points of φ are the size-1 conjugacy classes. These are the elements in the 3 3 center: Z(D6) = feg [ fr g = hr i. By the Orbit-Stabilizer theorem: jD j 12 j Stab(x)j = 6 = : j Orb(x)j j clG (x)j The stabilizer subgroups are as follows: 3 Stab(e) = Stab(r ) = D6, 2 4 5 Stab(r) = Stab(r ) = Stab(r ) = Stab(r ) = hri = C6, Stab(f ) = fe; r 3; f ; r 3f g = hr 3; f i, Stab(rf ) = fe; r 3; rf ; r 4f g = hr 3; rf i, Stab(r i f ) = fe; r 3; r i f ; r i f g = hr 3; r i f i. Sec 5.3 Examples of group actions Abstract Algebra I 6 / 10 (Not covered) Example 6: Groups acting on subgroups by conjugation Let G = D3, and let S be the set of proper subgroups of G: n o S = hei; hri; hf i; hrf i; hr 2f i : There is a right group action of D3 = hr; f i on S by conjugation: −1 τ : D3 −! Perm(S) ; τ(g) is the permutation that sends each H to g Hg. τ(e) = hei hri hf i hrf i 2 hr f i hei hf i τ(r) = hei hri hf i hrf i hr 2f i hri hr 2f i hrf i τ(r 2) = hei hri hf i hrf i hr 2f i The action diagram. τ(f ) = hei hri hf i hrf i hr 2f i Stab(hei) = Stab(hri) = D3 = ND3 (hri) 2 τ(rf ) = hei hri hf i hrf i hr f i Stab(hf i) = hf i = ND3 (hf i), Stab(hrf i) = hrf i = ND3 (hrf i), τ(r 2f ) = hei hri hf i hrf i hr 2f i 2 2 2 Stab(hr f i) = hr f i = ND3 (hr f i). Sec 5.3 Examples of group actions Abstract Algebra I 7 / 10 (Not covered) Example 6: Groups acting on subgroups by conjugation More generally, any group G acts on its set S of subgroups by conjugation: φ: G −! Perm(S) ; φ(g) is the permutation that sends each H to g −1Hg. This is a right action, but there is an associated left action: H 7! gHg −1. Let H ≤ G be an element of S. The orbit of H consists of all conjugate subgroups: Orb(H) = fg −1Hg j g 2 Gg : The stabilizer of H is the normalizer of H in G: −1 Stab(H) = fg 2 G j g Hg = Hg = NG (H) : The fixed points of φ are precisely the normal subgroups of G: Fix(φ) = fH ≤ G j g −1Hg = H for all g 2 Gg : The kernel of this action is G iff every subgroup of G is normal. In this case, φ is the trivial homomorphism: pressing the g-button fixes (i.e., normalizes) every subgroup. Sec 5.3 Examples of group actions Abstract Algebra I 8 / 10 Example 7: Groups acting on cosets of H by right-multiplication Fix a subgroup H ≤ G. Then G acts on its right cosets by right-multiplication: φ: G −! Perm(S) ; φ(g) = the permutation that sends each Hx to Hxg. Let Hx be an element of S = G=H (the right cosets of H). There is only one orbit. For example, given two cosets Hx and Hy, φ(x −1y) sends Hx 7−! Hx(x −1y) = Hy: The stabilizer of Hx is the conjugate subgroup x −1Hx: Stab(Hx) = fg 2 G j Hxg = Hxg = fg 2 G j Hxgx −1 = Hg = x −1Hx : Assuming H 6= G, there are no fixed points of φ. The only orbit has size [G : H] > 1. The kernel of this action is the intersection of all conjugate subgroups of H: \ −1 Ker φ = x Hx x2G Notice that hei ≤ Ker φ ≤ H, and Ker φ = H iff H E G. Sec 5.3 Examples of group actions Abstract Algebra I 9 / 10 Example 7a: S3 acting on cosets of H := h(12)i by right multiplication Sec 5.3 Examples of group actions Abstract Algebra I 10 / 10.