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MATH 611 NOTES GROUPS ACTING ON SETS OR G–SETS William A. Lampe May 26, 2009 The notion of a group acting on a set is an old one. The groups studied by Galois consisted of groups of permutations of roots of a polynomial. The notion of a group acting on a set is a slightly more general notion. We will study this notion from the point of view of universal algebra. Some of the associated notions are clearer this way. Definition. Let G = (cid:104)G; ·,−1,1(cid:105) be a group. A G–set is a unary algebra A = (cid:104)A;O(cid:105) where O = (g : g ∈ G) and 1 = ι and g◦h = g ·h for all g,h ∈ G. Often G A we will use the more suggestive notation A = (cid:104)A;G(cid:105) or A = (cid:104)A;G(cid:105) for a G–set. So given a group G, the class of all G–sets is a variety. Proposition. If G is a group and A = (cid:104)A;G(cid:105) is a G–set, then: (1) for each g ∈ G, the mapping g : A −→ A is a bijection (or permutation); (2) the mapping which sends g −→ g is a homomorphism from G onto (cid:104)G;◦,−1,ι (cid:105). A Definition. WhenGisagroupandA = (cid:104)A;G(cid:105)isaG–setandthehomomorphism of the above Proposition is 1–1, we say that the action of G on A is effective or faithful. Examples. Suppose G = (cid:104)G; ·,−1,1(cid:105) is a group. (1) Cayley’s representation or “G acting on itself by left multiplication” – (cid:104)G;(λ : g ∈ G)(cid:105) is a G–set. (Recall that λ (x) = g ·x for all x ∈ G.) g g (2) G acting on itself by conjugation – for g ∈ G the mapping τ : G −→ G is g defined by τ (x) = gxg−1. (cid:104)G;(τ : g ∈ G)(cid:105) is a G–set. g g (3) G acting on Sub(G) by conjugation – For g ∈ G and H ∈ Sub(G) we let τˆ (H) = gHg−1. Then (cid:104)Sub(G);(τˆ : g ∈ G)(cid:105) is a G–set. g g (4) G acting on G/H by left multiplication – Let H be a subgroup of G. We take G/H to be the set of left cosets of G by H; that is, G/H = {aH : ˆ ˆ a ∈ G}. For g ∈ G we define the function λ on G/H by λ (aH) = (ga)H. g g ˆ Then (cid:104)G/H;(λ : g ∈ G)(cid:105) is a G–set. g (5) S – If G = S , the full symmetric group on n, then (cid:104)n;S (cid:105) is a G–set. n n n 1 Cayley’s representation gives a faithful (or effective) action of G on itself, while “G acting on itself by conjugation” is a faithful action iff the center of G is {1}. ˆ By an abuse of notation, we will refer to the G–set (cid:104)G/H;(λ : g ∈ G)(cid:105) in (4) g as G/H. Proposition on Reducts. Suppose that G = (cid:104)G; ·,−1,1(cid:105) is a group and H is a subgroup of G and A = (cid:104)A;(g;g ∈ G)(cid:105) is a G–set. Then A = (cid:104)A;(g;g ∈ H)(cid:105) = (cid:104)A;H(cid:105) is an H–set (where H = (cid:104)H; ·,−1,1(cid:105)). So each of the above examples produces more examples by applying this Propo- sition. Exercises Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set. (1) If a ∈ A, then the subalgebra generated by a (= [a]) = {g(a) : g ∈ G}. (2) If a,b ∈ A and b ∈ [a], then [b] = [a]. (3) The one generated subalgebras of A form a partition of A. (4) Suppose A = B ∪ C and B and C are distinct one generated subalgebras (cid:12) of A. (We let B = (cid:104)B;(g(cid:12) : g ∈ G)(cid:105) and similarly for C.) If Φ ∈ Con(B) B and Ψ ∈ Con(C), then Φ ∪ Ψ ∈ Con(A). (5) State and prove a generalization of (4). Definitions. Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set. The one generated subalgebras of A are called orbits or the orbits of A or orbits of the action of G on A. If A has only one orbit, then A is said to be transitive, or the action of G on A is said to be transitive, or G is said to be a transitive permutation group. Let σ ∈ S , and consider the S –set (cid:104)n;S (cid:105). Then an orbit of σ is the same thing n n n as an orbit of the reduct (cid:104)n;[σ](cid:105), where [σ] denotes the subgroup of S generated n by σ. Definition. Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set and a ∈ A. Stab(a) = {g ∈ G : g(a) = a}. Stab(a) is called the stabilizer of a. Stabilizer Proposition 1. Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set and a ∈ A. Then Stab(a) is a subgroup of G. Stabilizer Proposition 2. Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set and a ∈ A and b = g(a). Then Stab(b) = g(Stab(a))g−1. That is, elements belonging to the same orbit have conjugate stabilizers. Some authors call isomorphic G–sets equivalent. Theorem 1. Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set. If A is transitive, then A is isomorphic to the G–set G/Stab(a) for any a ∈ A. 2 Corollary 1. Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set. If A is transitive, then |A| = [G : Stab(a)], the index of the stabilizer of a in G. Corollary 2. Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set. Then (cid:88) |A| = [G : Stab(a)] a∈R where R is a set containing exactly one element from each orbit. Corollary 3. Suppose G is a group. Then (cid:88) |G| = |C|+ [G : C(g)] g∈T where C denotes the center of G and C(g) denotes the centralizer of g (which = {x : xg = gx}) and T contains one element from each non trivial conjugacy class of G. The equation in Corollary 3 is called the class equation of G. Corollary 4. Let p be a prime number. Any finite group of prime power order has a center C (cid:54)= {1}. Theorem 2. Suppose G is a group and A = (cid:104)A;G(cid:105) is a G–set. If A is transitive, then Con(A) is isomorphic to (cid:104){H ∈ Sub(G) : H ⊇ Stab(a)};⊆(cid:105) for any a ∈ A. Some authors call a G–set primitive just in case it is simple. Recall that a simple algebra is one that has exactly two congruences. Corollary. Suppose G is a group and A = (cid:104)A;G(cid:105) is a transitive G–set. A is primitive iff for any a ∈ A, Stab(a) is a maximal subgroup of G. References Nathan Jacobson, Basic Algebra I, Second Edition, W. H. Freeman and Co., New York, 1985. Department of Mathematics University of Hawaii E-mail address: [email protected] 3