Let p be a prime and P be a finite p-group. We define a new characteristic subgroup K(P) of P and show that K(P) is a critical subgroup of P, which improves Thompson's proof for their existence. Secondly, we define several new characteristic subgroups K*(P) and K_i(P) for each positive integer i and show that they enjoy some interesting and new properties similar to those of critical subgroups. Lastly, for a solvable group G, we define a new characteristic subgroup Y(G), which satisfies the analogue of Thompson's result for solvable groups.