9b80133f-fed0-4a96-968c-8ad8d52ac45b20211228081145332naun:naunmdt@crossref.orgMDT DepositInternational Journal of Pure Mathematics2313-057110.46300/91019http://www.naun.org/cms.action?id=6985292021292021810.46300/91019.2021.8https://www.naun.org/cms.action?id=23293LiguoHeDept. of Math., Shenyang University of Technology Shenyang, 110870, PR ChinaYubingCaoDept. of Math., Shenyang University of Technology Shenyang, 110870, PR ChinaLet the finite group G act transitively and non-regularly on a finite set whose cardinality |Ω| is greater than one. Use N to denote the full set of fixed-point-free elements of G acting on along with the identity element. Write H to denote the stabilizer of some α ∈ Ω in G. In the note, it is proved that the subset N is a subgroup of G if and only if G is a Frobenius group. It is also proved G = {N}H, where {N} is the subgroup of G generated by N.102320211023202158-597https://www.naun.org/main/NAUN/puremath/2021/a142019-007(2021).pdf10.46300/91019.2021.8.7https://www.naun.org/main/NAUN/puremath/2021/a142019-007(2021).pdfR. Brown, Frobenius groups and classical maximal orders, Mem. Amer. Math. Soc., 2001, 717 D.G. Costanzo, M.L. Lewis, The cyclic graph of a 2- Frobenius group, arXive: 2103.15574v1[mathGR], 20 Mar 2021 The GAP Group, GAP — Groups, algorithms, and programming, Version 4.7.5, http://www.gapsystem.org, 2014 B. Huppert, Endliche Gruppen I, Springer–Verlag, Berlin-Heidelberg-New York, 1967 I.M. Isaacs, Character Theory of Finite Groups, Academic Press, New York, 1976 10.1016/j.jpaa.2005.11.005I. M. Isaacs, T. M. Keller, M.L. Lewis, Transitive permutation groups in which all derangements are involutions, Pure Appl. Algebra, 2006, 207: 717–724 H. Kurzweil, B. Stellmacher, The Theory of Finite Groups: an Introduction, Springer-Verlag New York, 2004 J. Maccrron, Frobenius groups with perfect order classes, arXive: 2103.00425v1[mathGR], 28 Feb 2021