Sylow Theorems

Question 1

What is the cauchy theorem in STs

Answer 1

Let G be a finite group of order p, if q divides |G| and is prime then G has an element of order q

Question 2

What is a p group

Answer 2

A group G is said to be a p-group if all elements of G have order of powers of a fixed prime p

Question 3

Order of a group

Answer 3

Number of elements in the group

Question 3

Order of a group

Answer 3

Number of elements in the group

Question 4

Order of an element g of a group

Answer 4

The smallest positive inter n, such that g^n=e

Question 5

Define p-subgroup

Answer 5

Let G be a group and let H be subgroup of G which is a p-group, then H is the p-subgroup of G. In particular the identity Is a p-subgroup of every prime p ||=1 =p^0

Question 6

First Sylow Theorem

Answer 6

let G be a group of order p^n(m) with n=>1,p prime and gcd (p,m)=1.Then G contains a subgroup of order p^i for each 1<= i =>n and every subgroup of G of order p^i (n<1) is normal in some subgroup of order p^(1+i)