GROUP ACTIONS Abstract. Monoid and group actions. Homogeneous spaces and conjugacy. Structure Theorem for G-sets. Restriction and induc- tion. Wreath products. 1. Representations 1.1. Homomorphisms. Definition 1.1. Consider a function f : G ! H. (a) If (G; ·) and (H; ·) are semigroups, the function f is a semigroup homomorphism if 8 x; y 2 G ; xf · yf = (xy)f (i.e., f preserves the multiplication). (b) If (G; ·; 1) and (H; ·; 1) are monoids, a semigroup homomor- phism f is a monoid homomorphism if 1f = 1 (i.e., f preserves the identity). (c) If (G; ·; 1) and (H; ·; 1) are groups, the function f is a group homomorphism if 8 x 2 G; (xf)−1 = x−1f (i.e., f preserves the inversion). The large concrete categories of semigroup, monoid, and group ho- momorphisms are respectively denoted Sgp, Mon, and Gp. Remark 1.2. More generally, a function between algebraic structures is a homomorphism if it preserves those structures. For example, vector space homomorphisms are linear transformations. A bijective function between algebraic structures A and B is an isomorphism; one then ∼ writes A = B. The following lemma is very useful. Lemma 1.3. If a function f :(G; ·; 1) ! (H; ·; 1) between groups is a semigroup homomorphism, then it is a group homomorphism. 1 2 GROUP ACTIONS 1.1.1. The First Isomorphism Theorem for Sets. Theorem 1.6 below decomposes an arbitrary function f : X ! Y in the form f = sbj, with a surjection s, a bijection b, and an injection j. Definition 1.4. The (relation) kernel or relation kernel ker f of f is the equivalence relation on X defined by x1 ker f x2 , x1f = x2f for elements x1, x2 of the domain X of f. Remark 1.5. To distinguish from the relation kernel, the kernel sub- space of a linear transformation f is written as Ker f. Then a similar convention applies in other contexts (groups, rings, etc.). Theorem 1.6 (First Isomorphism Theorem for Sets). Let f : X ! Y be a function. With the surjection s = nat kerf : X ! Xker f ; x 7! xker f and the injection j : Xf ! Y ; xf 7! xf, there is a well-defined bijection (set isomorphism) b: Xker f ! Xf; xker f 7! xf such that f = sbj. / x_ xfO f / X YO s j Xker f / Xf b _ xker f / xf 1.1.2. The First Isomorphism Theorem for Algebras. If f : X ! Y is a homomorphism of algebraic structures, a refined version of Theorem 1.6 applies. Theorem 1.7 (First Isomorphism Theorem for Algebras). Suppose that f : X ! Y be a homomorphism. The surjective homomorphism s = nat kerf : X ! Xker f ; x 7! xker f and the injective homomorphism j : Xf ! Y ; xf 7! xf compose with a well-defined algebra isomorphism b: Xker f ! Xf; xker f 7! xf to yield f = sbj. Remark 1.8. Applied to a linear transformation f : X ! Y between finite-dimensional real vector spaces, Theorem 1.7 implies the relation dim X = dim Ker f + dim Xf (Exercise 3). GROUP ACTIONS 3 1.2. Semigroup representations. Let X be a set. Recall the semi- group of functions (XX ; ·) under composition. Definition 1.9 (Semigroup representations). Let M be a semigroup. (a) A homomorphism r : M ! XX ; m 7! mr is called a (right) (semigroup) representation of M (on X). (b) A right representation r is faithful( ) if the function r is injective. (c) A homomorphism l : M ! XX op ; m 7! ml is called a (left) (semigroup) representation of M (on X). (d) A left representation l is faithful if the function l is injective. Proposition 1.10 (Mixed associative laws). Let X be a set and let M be a semigroup. (a) Consider a right representation r : M ! XX ; m 7! mr of M. Then the mixed associative law (1.1) 8 x 2 X; 8 m; n 2 M; (xmr)nr = x(mn)r holds. (b) Consider a left representation l : M ! XX ; m 7! ml of M. Then the mixed associative law (1.2) 8 x 2 X; 8 m; n 2 M; (ml ◦ nl)(x) = (mn)l(x) holds. (c) A function r : M ! XX is a right representation if the mixed associative law (1.1) holds. (d) A function l : M ! XX is a left representation if the mixed associative law (1.2) holds. Proposition 1.10 shows how algebraic and Euler function notations are natural in their respectively dual contexts. Using the \wrong" notation for the context inserts extra twists in the mixed associative laws. 1.2.1. Currying. A right representation of a semigroup M on a set X has been defined as a semigroup homomorphism r : M ! XX . As an element of Set(M; Set(X; X)), the function r may be Curried to ρ: X × M ! X;(x; m) 7! xmr : The mixed associative law (1.1) encodes as the commutative diagram ρ×1 (1.3) X × M × M M / X × M 1 ×∇ ρ X × / X M ρ X 4 GROUP ACTIONS in the category Set, with r: M × M ! M;(m; n) 7! mn as the semigroup multiplication. 1.3. Some examples. Example 1.11 (Homotheties). Let U be a real vector space. Then there is a representation (R; ·) ! U U , under which a real number (\scalar") λ is represented by the homothety that multiplies each vector in U by the constant scale factor λ. Definition 1.12. Let s be an element of a magma (S; ·). (a) The right multiplication by s is the mapping R(s): S ! S; x 7! x · s in SS. (b) The left multiplication by s is the mapping L(s): S ! S; x 7! s · x in SS. Proposition 1.13. Suppose that (S; ·) is a semigroup. Then there is a right representation R: S ! SS; s 7! R(s) of S on S. Proof. The mixed associative law xR(s)R(t) = xR(st), for elements x; s; t of S, follows directly from the associative law (xs)t = x(st). Definition 1.14. The representation R of Proposition 1.13 is known as the right regular representation of the semigroup (S; ·). 1.3.1. Induced representations. Definition 1.15. Let f : M ! N be a semigroup homomorphism. Let r : N ! XX be a right representation of N on a set X. The composite homomorphism fr : M ! XX is called the (right) representation of M on X induced by f. Example 1.16 (The flip-flop). Consider the set 2 = f0; 1g of bits. The set S = fid2; c0; c1g, with constant or reset functions ci : 2 ! 2 having image fig for i 2 2, forms a semigroup (Exercise 9). The right representation of S induced by the inclusion S,! 22; f 7! f is called the flip-flop. Along with right regular representations of finite simple groups, it is one of the key components in the Krohn-Rhodes theory GROUP ACTIONS 5 analysing finite semigroups, as a generalization of the Jordan-H¨older decomposition for groups. 1.4. Monoid and group actions. If X is a set and M is a monoid, then (right or left) monoid representations of M on X are defined by monoid homomorphisms M ! XX or M ! (XX )op in place of the semigroup homomorphisms of x1.2. Proposition 1.10(c)(d) has its corresponding analogue: Proposition 1.17. Let X be a set and let M be a monoid. (a) A function r : M ! XX is a right monoid representation if the mixed associative law (1.1) holds, and if x1r = x for all x 2 X. (b) A function l : M ! XX is a left monoid representation if the mixed associative law (1.2) holds, and if 1l(x) = x for all x 2 X. Definition 1.18. Let X be a set, and let M be a group. (a) A group homomorphism r : M ! X!; m 7! mr is called a (right) permutation representation of M (on X). (c) A group homomorphism l : M ! (X!)op ; m 7! ml is called a (left) permutation) representation of M (on X). Lemma 1.3 implies the following. Proposition 1.19. Let X be a set and let M be a group. (a) A function r : M ! X! is a right permutation representation if the mixed associative law (1.1) holds. (b) A function l : M ! X! is a left permutation representation if the mixed associative law (1.2) holds. Theorem 1.20 (Cayley's Theorem). Let (G; ·; 1) be a group. Then there is a faithful right permutation representation R: G ! G!; g 7! R(g) of G on G. Definition 1.21. The representation R of Theorem 1.20 is known as the right regular permutation representation of the group (G; ·; 1). 1.5. Dual representations. Proposition 1.22. Let X and Y be sets. Let M be a semigroup. Let r : M ! XX be a right representation of M on X. Then x(mlf) = (xmr)f ; for x 2 X, m 2 M, and f : X ! Y , defines a left representation l of M on Y X . 6 GROUP ACTIONS Proof. By Proposition 1.10, the mixed associative law (1.4) (ml ◦ nl)(f) = (mn)lf must be verified for f 2 Y X and m; n 2 M. The functions on each side of (1.4) have the same domain X and codomain Y . Then for an element x of X, one has ( ( )) x(ml ◦ nl)(f) = x ml nlf = (xmr)(nlf) ( ) ( ) = xmrnrf = x(mn)r f = x (mn)lf ; so the two sides of (1.4) represent equal functions.