Page 83 - Ellingham, Mark, Mariusz Meszka, Primož Moravec, Enes Pasalic, 2014. 2014 PhD Summer School in Discrete Mathematics. Koper: University of Primorska Press. Famnit Lectures, 3.
P. 83
mož Moravec: Some Topics in the Theory of Finite Groups 71
Corollary 3.2.20 (Class equation) Let G be a finite group and let x1, . . . , xr be the repre-
sentatives of conjugacy classes of non-central elements of G . Then
r
|G | = |Z (G )| + |G : CG (xi )|.
i =1
For g ∈ G denote by fix(g ) the number of fixed points of g (considered as an element
of Sym X ). We have:
Theorem 3.2.21 (Orbit-counting Lemma) Let a finite group G act on a set X . Then
|X /G | = 1 fix(g ).
|G | g ∈G
PROOF. We will count the pairs (x , g ) ∈ X × G with the property that x g = x ; let us call
these pairs good pairs. On one hand, a given g ∈ G is a member of fix(g ) good pairs,
hence the total number of good pairs is g ∈G fix(g ). On the other hand, x ∈ X is a mem-
ber of | stabG (x )| good pairs. The orbit of x thus produces | orbG (x )| · | stabG (x )| = |G | good
pairs, hence there are |X /G | · |G | good pairs in total. We get the result.
Sylow theorems
Since the action of G on itself by right multiplication is faithful, we have that the corre-
sponding homomorphism G → SymG is injective. In particular, we have:
Theorem 3.2.22 (Cayley’s theorem) Every finite group is isomorphic to a subgroup of Sn
for some positive integer n.
Another classical result that can be proved using actions is Cauchy’s theorem which
provides a basis for Sylow theorems. It goes as follows:
Theorem 3.2.23 (Cauchy’s theorem) Let G be a finite group. If a prime p divides |G |,
then G contains an element of order p .
Theorem 3.2.24 (Sylow’s theorem) Let G be a group of order p a · m , where m is not di-
visible by the prime p . Then the following holds:
1. G contains at least one subgroup of order p a . Any two subgroups of this order are
conjugate in G . They are called the Sylow p -subgroups of G .
2. For each n ≤ a , G contains at least one subgroup of order p n . Every such subgroup
is contained in a Sylow p -subgroup.
3. Let sp be the number of Sylow p -subgroups of G . Then sp ≡ 1 mod p and sp divides
m.
Corollary 3.2.20 (Class equation) Let G be a finite group and let x1, . . . , xr be the repre-
sentatives of conjugacy classes of non-central elements of G . Then
r
|G | = |Z (G )| + |G : CG (xi )|.
i =1
For g ∈ G denote by fix(g ) the number of fixed points of g (considered as an element
of Sym X ). We have:
Theorem 3.2.21 (Orbit-counting Lemma) Let a finite group G act on a set X . Then
|X /G | = 1 fix(g ).
|G | g ∈G
PROOF. We will count the pairs (x , g ) ∈ X × G with the property that x g = x ; let us call
these pairs good pairs. On one hand, a given g ∈ G is a member of fix(g ) good pairs,
hence the total number of good pairs is g ∈G fix(g ). On the other hand, x ∈ X is a mem-
ber of | stabG (x )| good pairs. The orbit of x thus produces | orbG (x )| · | stabG (x )| = |G | good
pairs, hence there are |X /G | · |G | good pairs in total. We get the result.
Sylow theorems
Since the action of G on itself by right multiplication is faithful, we have that the corre-
sponding homomorphism G → SymG is injective. In particular, we have:
Theorem 3.2.22 (Cayley’s theorem) Every finite group is isomorphic to a subgroup of Sn
for some positive integer n.
Another classical result that can be proved using actions is Cauchy’s theorem which
provides a basis for Sylow theorems. It goes as follows:
Theorem 3.2.23 (Cauchy’s theorem) Let G be a finite group. If a prime p divides |G |,
then G contains an element of order p .
Theorem 3.2.24 (Sylow’s theorem) Let G be a group of order p a · m , where m is not di-
visible by the prime p . Then the following holds:
1. G contains at least one subgroup of order p a . Any two subgroups of this order are
conjugate in G . They are called the Sylow p -subgroups of G .
2. For each n ≤ a , G contains at least one subgroup of order p n . Every such subgroup
is contained in a Sylow p -subgroup.
3. Let sp be the number of Sylow p -subgroups of G . Then sp ≡ 1 mod p and sp divides
m.
