
theorem Th14:
  for G being finite commutative Group,
  m be Nat,
  A be Subset of G
  st A ={x where x is Element of G: x|^m = 1_G} holds
  ex H being strict finite Subgroup of G
  st the carrier of H = A & H is commutative normal
  proof
    let G be finite commutative Group,
    m be Nat,
    A be Subset of G;
    assume A ={x where x is Element of G: x|^m = 1_G};
    then
    A <> {} & (for g1,g2 be Element of G
    st g1 in A & g2 in A holds g1 * g2 in A) &
    for g be Element of G st g in A holds g" in A by Th13; then
    consider H being strict Subgroup of G such that
    A1: the carrier of H = A by GROUP_2:52;
    H is normal by GROUP_3:116;
    hence thesis by A1;
  end;
