1.3 Sylow’s Theorem

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In this section all groups are finite. Let G be one such. If p is a prime

number, and if n is a nonnegative integer with pn||G|, pn+1 6 | |G|, write

pn = |G|p, and call pn the p-part of |G|. If |G|p = pn, and if P _ G

with |P| = pn, call P a p-Sylow subgroup of G. The set of all p-Sylow

subgroups of G is denoted Sylp(G). Sylow’s Theorem (see Theorem 1.3.2,

below) provides us with valuable information about Sylp(G); in particular,

that Sylp(G) 6= ;, thereby providing a “partial converse” to Lagrange’s

Theorem (Theorem 1.1.1, above). First a technical lemma3

Lemma 1.3.1 Let X be a finite set acted on by the finite group G, and let

p be a prime divisor of |G|. Assume that for each x 2 X there exists a

p-subgroup P(x) _ G with {x} = Fix(P(x)). Then

(1) G is transitive on X, and

(2) |X| _ 1(mod p).

Here it is:

Theorem 1.3.2 (Sylow’s Theorem) Let G be a finite group and let p be

a prime.

(Existence) Sylp(G) 6= ;.

(Conjugacy) G acts transitively on Sylp(G) via conjugation.

(Enumeration) |Sylp(G)| _ 1(mod p).

(Covering) Every p-subgroup of G is contained in some p-Sylow subgroup

of G.

Exercises 1.3

1. Show that a finite group of order 20 has a normal 5-Sylow subgroup.

2. Let G be a group of order 56. Prove that either G has a normal 2-Sylow

subgroup or a normal 7-Sylow subgroup.

3See, M. Aschbacher, Finite Group Theory, Cambridge studies in advanced mathematics

10, Cambridge University Press 1986.

1.3. SYLOW’S THEOREM 13

3. Let |G| = pem, p > m, where p is prime. Show that G has a normal

p-Sylow subgroup.

4. Let |G| = pq, where p and q are primes. Prove that G has a normal

p-Sylow subgroup or a normal q-Sylow subgroup.

5. Let |G| = pq2, where p and q are distinct primes. Prove that one of

the following holds:

(1) q > p and G has a normal q-Sylow subgroup.

(2) p > q and G has a normal p-Sylow subgroup.

(3) |G| = 12 and G has a normal 2-Sylow subgroup.

6. Let G be a finite group and let N /G. Assume that for all e 6= n 2 N,

CG(n) _ N. Prove that (|N|, [G : N]) = 1.

7. Let G be a finite group acting transitively on the set X. Let x 2

X, Gx = StabG(x), and let P 2 Sylp(Gx). Prove that NG(P) acts

transitively on Fix(P).

8. (The Frattini argument) Let H /G and let P 2 Sylp(G), with P _ H.

Prove that G = HNG(P).

9. The group G is called a CA-group if for every e 6= x 2 G, CG(x) is

abelian. Prove that if G is a CA-group, then

(i) The relation x _ y if and only if xy = yx is an equivalence relation

on G#;

(ii) If C is an equivalence class in G#, then H = {e}[C is a subgroup

of G;

(iii) If G is a finite group, and if H is a subgroup constructed as in

(ii) above, then (|H|, [G : H]) = 1. (Hint: If the prime p divides

the order of H, show that H contains a full p-Sylow subgroup of

G.)

14 CHAPTER 1. GROUP THEORY

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