Sylow's theorems

Question 1

p prime, |G| = p^r m then,

Answer 1

i) G contains at least one sylow p subgroup where |p| = p^r
ii) subgroup of G of order p^r form a conjugacy class
iii) If X subgroup of G, X a pgroup, then there exists an x st X is a subgroup of p^x
iv) If n is the number of subgroups of G of order p^r, then n = 1(modp)

Question 2

Sylp G denotes

Answer 2

denotes the set of all sylow p subgroups of G

Question 3

np

Answer 3

np=[G: NG(p)]

np doesnt divide m

Question 4

If |G|= pq, p and q are distinct primes, such that p does not divide q-1, then….

Answer 4

G has a normal sylow p subgroup

Question 5

If |G|= pq, p and q are distinct primes, such that p does not divide q-1 and p

Answer 5

then G is cyclic

Question 6

If G is a group, |G|=pqr where p,q,r are distinct primes,

Answer 6

then G is not simple