5. Sylow's Theorem

Question 1

Sylow’s First Theorem

Answer 1

Let p be a prime number. Let G be a finite group of order pam where a≥1 and m∈N such that p∤ m. Then G contains a subgroup of order pr for each r ≤ a.

Question 2

When are Sylow p-subgroups always conjugate.

Answer 2

Let p be a prime number. Let G be a finite group such that p | |G|

Question 3

State sylows 3rd theorem about p-subgroups

Answer 3

Let p be a prime number. Let G be a finite group of order pam
where a ≥ 1 and m ∈ N such that p does not divide m. Let np be
the number of Sylow p-subgroups of G. Then np∣∣m and np ≡ 1 modp