Suppose that is a finite group and is a subgroup of . The subgroup is said to be weakly-supplemented in if there exists a proper subgroup of such that . In this note, by using the weakly-supplemented subgroups, we point out several mistakes in the proof of Theorem 1.2 of Q. Zhou (2019) and give a counterexample.
A subgroup of a finite group is weakly-supplemented in if there exists a proper subgroup of such that . In the paper, we extend one main result of Kong and Liu (2014).
An important theorem by J. G. Thompson says that a finite group is -nilpotent if the prime divides all degrees (larger than 1) of irreducible characters of . Unlike many other cases, this theorem does not allow a similar statement for conjugacy classes. For we construct solvable groups of arbitrary -lenght, in which the lenght of any conjugacy class of non central elements is divisible by .
Let be a fixed positive integer. In this paper, we consider finite groups each of whose nonlinear character degrees has exactly prime divisors. We show that such groups are solvable whenever . Moreover, we prove that if is a non-solvable group with this property, then and is an extension of or by a solvable group.
In this paper we describe some algorithms to identify permutable and Sylow-permutable subgroups of finite groups, Dedekind and Iwasawa finite groups, and finite T-groups (groups in which normality is transitive), PT-groups (groups in which permutability is transitive), and PST-groups (groups in which Sylow permutability is transitive). These algorithms have been implemented in a package for the computer algebra system GAP.