1 ·································  p is prime
           2 ·································  pow(P,i) =p*p...p
           3 ·························  if a>0, b>0 => p | pow(p,a)
           4 ·························  p|s and p|t => p | (s+t)
           5 ·································  |G|= pow(P,a)*m
           6 ·································  |P|= pow(P,a)
           7 ·································  |G|/|P|= |G/P|
           8 ·································  partitions of a set donšt intersect
           9 ·································  orbits are partitions of a set
          10 ·································  sum(parts)=whole if parts donšt intersec
          11 ·························  p does not divide m
          12 ·················  size of orbit =1 => orbit consists of one fixed element
          13 ·················  cancellation laws in Group
          14 ·············  if P < G, gP=P => g is a member of P
          15 ·········  for all a in A, a is member of B => A is a subset of B
          16 ·····································  Q < G
          17 ·····································  G acts on G/P
          18 ·····································  if a group acts on a set, then a subgroup acts on the set
          19 ·············  syl(p,G)={P < G: |P|=pow(p,a)}
          20 ·············  Q is a member of syl(p,G)
          21 ·············  P is a member of syl(p,G)
          22 ·································  size of orbit divides size of acting group
          23 ·········  |P|=|x*P*pow(x,-1)|
   1+  2= 24 ·····························  Only pow(p,i) divides pow(p,a)
  16+ 17+ 18
        = 28 ·································  Q acts on G/P
  19+ 20+ 21
        = 29 ·········  |Q|=|P|
1 22+ 28= 32 ·····························  |O| divides pow(p,a)
  23+ 29= 33 ·····  |Q|=|x*P*pow(x,-1)|
  24+ 32= 34 ·························  |O|=pow(p,i) and i>=0
  33+ 43= 44 ··  there exists x in G s.t. x*P*pow(x,-1)=q
      44= 45 let |G| = pow(p,a)*m for a>=1, p prime and p does not divide m. Set Syl(p,G)={P< G: |P|=pow(p,a}. P, Q are member of Syl(p,G), Then there exists x in G s.t. x*P*pow(x,-1)=Q