Sylow's theorem

Question 1

What is Sylow’s first theorem?

Answer 1

Let G be a finite group. If |G| = P^km where P € Z and P is prime; k € Z and K ≥ 0; m € N, and P does not divide m. There exists at least one subgroup in G of order P^k.

Question 2

What is Sylow’s seconds theorem?

Answer 2

Let G be a finite group. If |G| = P^km where P € Z and P is prime; k € Z and K ≥ 0; m € N, and P does not divide m. If H and K are members of the same Sylow p-subgroup of G, then there exists g € G: H = gKg^-1. That is, H and K are conjugates of each other.

Question 3

State Sylow’s third theorem.

Answer 3

Let G be a finite group. If |G| = P^km where P € Z and P is prime; k € Z and K ≥ 0; m € N, and P does not divide m. Then the number of a Sylow P-subgroup, n_p, is
n_p ≡ 1(mod p), and n_p | m

Question 4

What is a Sylow P-subgroup?

Answer 4

Let G be a finite group. If |G| = P^km where P € Z and P is prime; k € Z and K ≥ 0; m € N, and P does not divide m.
If there is a subgroup H in G of order P^k, then this subgroup is a Sylow subgroup.

Question 5

What is a simple group?

Answer 5

A simple group is a group whose only normal subgroups are the identity subgroup (of order 1) and the group itself. I.e., whose only normal subgroups are the trivial subgroups.