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**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

where n 1 (mod p) but n^{p}= e, b^{q}= e, ba = a^{n}b^{q}1 (mod p).

**13.**
By direct computation, find the number of Sylow 3-subgroups
and the number of Sylow 5-subgroups of the symmetric group S_{5}.
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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