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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:
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 }
is called the orbit of x under G.G_{x} = { a in G | ax = x }
is called the stabilizer of x in G.S^{G} = { x in S | ax = x for all a in G }
is called the subset of S fixed by G.
Proposition 7.3.4.
Let G be a group that acts on the set S, and let
x be an element of S.
Proposition 7.3.5.
Let G be a finite group acting on the set S.
Theorem 7.3.6.
Let G be a finite group acting on the finite set S. Then
| S | = | S^{G} | + _{} [ G : G_{x} ],
where is a set of representatives of the orbits Gx for which | Gx | > 1.
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.
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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