reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem
  for G being finite Group
  for N being strict normal Subgroup of G
  st card N, index N are_coprime
  holds N is characteristic Subgroup of G
proof
  let G be finite Group;
  let N be strict normal Subgroup of G;
  assume A1: card N, index N are_coprime;
  consider m being Nat such that
  A2: m = card N;
  consider n being Nat such that
  A3: n = index N;
  A4: card G = m*n by A2,A3,GROUP_2:147;
  for phi being Automorphism of G holds Image(phi|N) = N
  proof
    let phi be Automorphism of G;
    set K = Image(phi|N);
    reconsider K as strict normal Subgroup of G by Th49;
    K = phi .: N by GRSOLV_1:def 3;
    then B1: card K = card N by Th19,GROUP_6:73;
    set d = card(N /\ K);
    d divides m
    proof
      N /\ K is Subgroup of N by GROUP_2:88;
      hence thesis by A2,GROUP_2:148;
    end;
    then consider q being Nat such that
    B2: m = d*q by NAT_D:def 3;
    B3: q<>0 by A2,B2;
    card(N "\/" K) = m*q
    proof
      K /\ N = N /\ K
      proof
        carr(K /\ N) = (carr K) /\ (carr N) by GROUP_2:def 10
                    .= carr(N /\ K) by GROUP_2:def 10;
        hence thesis by GROUP_2:59;
      end;
      then d*card(N "\/" K) = m*(d*q) by A2,B1,B2,Th70
                           .= d*(m*q);
      hence card(N "\/" K) = m*q by XCMPLX_1:5;
    end;
    then q divides n by A2,A4,Th67a,GROUP_2:148;
    then q=1 by A1,A2,A3,B2,B3,Th68;
    hence Image(phi|N) = N by A2,B1,B2,Th66;
  end;
  hence N is characteristic Subgroup of G by Def3;
end;
