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 Th50:
  for G being Group
  for H being strict Subgroup of G
  holds H is characteristic iff
        (for phi being Automorphism of G
         for x being Element of G
         st x in H
         holds phi.x in H)
proof
  let G be Group;
  let H be strict Subgroup of G;
  thus H is characteristic implies (for phi being Automorphism of G
                                    for x being Element of G
                                    st x in H
                                    holds phi.x in H)
  proof
    assume B1: H is characteristic;
    let phi be Automorphism of G;
    let x be Element of G;
    assume B2: x in H;
    B3: H = Image(phi|H) by B1
         .= phi .: H by GRSOLV_1:def 3;
    dom phi = the carrier of G by FUNCT_2:def 1;
    then phi.x in phi .: (the carrier of H) by B2,FUNCT_1:def 6;
    hence thesis by B3,GRSOLV_1:8;
  end;
  thus (for phi being Automorphism of G
        for x being Element of G
        st x in H
        holds phi.x in H)
       implies H is characteristic
  proof
    assume B1: for phi being Automorphism of G
               for x being Element of G st x in H holds phi.x in H;
    for phi being Automorphism of G holds Image(phi|H) is Subgroup of H
    proof
      let phi be Automorphism of G;
      for x being Element of H holds phi.x in H
      proof
        let x be Element of H;
        reconsider g=x as Element of G by GROUP_2:42;
        g in H;
        hence thesis by B1;
      end;
      hence Image(phi|H) is Subgroup of H by Th44;
    end;
    hence H is characteristic by Th40;
  end;
end;
