Applications of Commutator-Type Operators to $p$-Groups

For a p-group G admitting an automorphism $\varphi$ of order $p^n$ with exactly $p^m$ fixed points such that ${\varphi }^{p^{n-1}}$ has exactly $p^k$ fixed points, we prove that G has a fully-invariant subgroup of m-bounded nilpotency class with $(p,n,m,k)$-bounded index in G. We also establish its analogue for Lie p-rings. The proofs make use of the theory of commutator-type operators.