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§ 7.3 Groups acting on sets

 
Definition 7.3.1. Let G be a group and let S be a set. A multiplication of elements of S by elements of G (defined by a function from G × S -> S) is called a group action of G on S provided for each element x in S:

(i) a(bx) = (ab)x for all a,b in G, and

(ii) ex = x for the identity element e of G.

 
Proposition 7.3.2. Let G be a group and let S be a set. Any group homomorphism from G into the group Sym(S) of all permutations of S defines an action of G on S. Conversely, every action of G on S arises in this way.

 
Definition 7.3.3. Let G be a group acting on the set S. For each element x in S, the set

Gx = { s in S | s=ax   for some   a in G }

G_x = { a in G | ax = x }

S^G = { x in S | ax = x   for all   a in G }

 
Proposition 7.3.4. Let G be a group that acts on the set S, and let x be an element of S.

(a) The stabilizer G_x of x in G is a subgroup of G.

(b) There is a one-to-one correspondence between the elements of the orbit Gx of x under G and the left cosets of G_x in G.

 
Proposition 7.3.5. Let G be a finite group acting on the set S.

(a) The orbits of S (under the action of G) partition S.

(b) For any x in S, | Gx | = [ G : G_x ].

 
Theorem 7.3.6. Let G be a finite group acting on the finite set S. Then

| S | = | S^G | + [ G : G_x ],

 
Lemma 7.3.7. Let G be a finite p-group acting on the finite set S. Then

| S | | S^G | (mod p).

 
Theorem 7.3.8. [Cauchy] If G is a finite group and p is a prime divisor of |G|, then the number of solutions in G of the equation x^p = e is a multiple of p. In particular, G has an element of order p.

§ 7.3 Groups acting on sets: Solved problems

 
11. Let G be a group which has a subgroup of index 6. Prove that G has a normal subgroup whose index is a divisor of 720.     Solution

12. Let G act on the subgroup H by conjugation, let S be the set of all conjugates of H, and let µ : G -> Sym (S) be the corresponding homomorphism. Show that ker(µ) is the intersection of the normalizers N(aHa ^-1) of all conjugates.     Solution

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