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§ 7.4 The Sylow theorems

 
Theorem 7.4.1. [First Sylow Theorem] Let G be a finite group. If p is a prime such that p^k is a divisor of |G| for some k 0, then G contains a subgroup of order p^k.

 
Definition 7.4.2. Let G be a finite group, and let p be a prime number. A subgroup P of G is called a Sylow p-subgroup of G if |P| = p^k for some integer k 1 such that p^k is a divisor of |G| but p^k+1 is not.

 
Lemma 7.4.3. Let G be a finite group with |G| = mp^k, where k 1 and m is not divisible by p. If P is a normal Sylow p-subgroup, then P contains every p-subgroup of G.

 
Theorem 7.4.4. [Second and Third Sylow Theorems] Let G be a finite group of order n, and let p be a prime number.

(a) All Sylow p-subgroups of G are conjugate, and any p-subgroup of G is contained in a Sylow p-subgroup.

(b) Let n = mp^k, with gcd(m,p)=1, and let s be the number of Sylow p-subgroups of G.
Then s | m and s 1 (mod p).

 
Proposition 7.4.5. Let p > 2 be a prime, and let G be a group of order 2p. Then G is either cyclic or isomorphic to the dihedral group D_p of order 2p.

 
Proposition 7.4.6. Let G be a group of order pq, where p > q are primes.

(a) If q is not a divisor of p-1, then G is cyclic.

(b) If q is a divisor of p-1, then either G is cyclic or else G is generated by two elements a and b satisfying the following equations:

a^p = e,     b^q = e,     ba = aⁿb

where n 1 (mod p) but n^q 1 (mod p).

§ 7.4 The Sylow Theorems: Solved problems

 
13. By direct computation, find the number of Sylow 3-subgroups and the number of Sylow 5-subgroups of the symmetric group S₅. Check that your calculations are consistent with the Sylow theorems.     Solution

14. How many elements of order 7 are there in a simple group of order 168?     Solution

15. Prove that a group of order 48 must have a normal subgroup of order 8 or 16.     Solution

16. Let G be a group of order 340. Prove that G has a normal cyclic subgroup of order 85     Solution

17. Show that there is no simple group of order 200.     Solution

18. Show that a group of order 108 has a normal subgroup of order 9 or 27.     Solution

19. If p is a prime number, find all Sylow p-subgroups of the symmetric group S_p.     Solution

20. Prove that if G is a group of order 56, then G has a normal Sylow 2-subgroup or a normal Sylow 7-subgroup.     Solution

21. Prove that if N is a normal subgroup of G that contains a Sylow p-subgroup of G, then the number of Sylow p-subgroups of N is the same as that of G.     Solution

22. Prove that if G is a group of order 105, then G has a normal Sylow 5-subgroup and a normal Sylow 7-subgroup.     Solution

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