reserve x,y,y1,y2 for set;
reserve G for Group;
reserve a,b,c,d,g,h for Element of G;
reserve A,B,C,D for Subset of G;
reserve H,H1,H2,H3 for Subgroup of G;
reserve n for Nat;
reserve i for Integer;
reserve L for Subset of Subgroups G;
reserve N2 for normal Subgroup of G;

theorem Th137:
  for G being strict Group, H being strict Subgroup of G holds H
  is normal Subgroup of G iff Normalizer H = G
proof
  let G be strict Group, H be strict Subgroup of G;
  thus H is normal Subgroup of G implies Normalizer H = G
  proof
    assume
A1: H is normal Subgroup of G;
    now
      let a be Element of G;
      H |^ a = H by A1,Def13;
      hence a in Normalizer H by Th134;
    end;
    hence thesis by GROUP_2:62;
  end;
  assume
A2: Normalizer H = G;
  H is normal
  proof
    let a be Element of G;
    a in Normalizer H by A2;
    then ex b being Element of G st b = a & H |^ b = H by Th134;
    hence thesis;
  end;
  hence thesis;
end;
