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We point out some gaps in the paper (Monatsh Math doi:10.1007/s00605-016-0877-1, 2016) by Kong and Guo, and give new brief proofs of the main results.

In this note, we give two new conditions for a finite group to be \(p\)-nilpotent under the assumption that all subgroups with an any fixed order are \(E\)-\(S\)-supplemented, and some recent results are generalized.

We point out an error in the paper ‘On an extension of the Frobenius’ theorem about p-nilpotency of a finite group’ (Monatsh Math 175(1):133–138, 2014) by Kong, and give a new brief proof of the main result.

In this note, we show that the main result in the paper ‘On an extension of the Frobenius’ theorem about \(p\)-nilpotency of a finite group’ (Monatsh Math doi:10.1007/s00605-013-0492-3, 2013) by Kong can be seen as a corollary of a theorem in the paper ‘On finite groups with prime-power order \(S\)-quasinormally embedded subgroups’ (Monatsh Math 162:329–339, 2011) by Wei and Guo.

We point out an error in the paper (Monatsh Math. doi: 10.1007/s00605-013-0492-3 , 2013) by Kang, and give new brief proofs of the main results.