reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem
  for G being finite Group, N being normal Subgroup of G holds
  N = center G & G./.N is cyclic implies G is commutative
proof
  let G be finite Group;
  let N be normal Subgroup of G;
  assume
A1: N = center G & G./.N is cyclic;
  then N is Subgroup of center G by GROUP_2:54;
  hence thesis by A1,Th10;
end;
