## Minimal involution-free nongroup reduced twisted subsets.(English. Russian original)Zbl 1232.20030

Math. Notes 88, No. 6, 860-867 (2010); translation from Mat. Zametki 88, No. 6, 902-910 (2010).
Summary: A subset $$K$$ of a group $$G$$ is said to be twisted if $$1\in K$$ and the element $$xy^{-1}x$$ lies in $$K$$ for any $$x,y\in K$$. We study finite involution-free twisted subsets that are not subgroups but all of whose proper twisted subsets are subgroups. We also study the groups generated by such twisted subsets.

### MSC:

 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20F05 Generators, relations, and presentations of groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks
Full Text:

### References:

 [1] A. L. Myl’nikov, ”Finite twisted groups,” Sibirsk. Mat. Zh. 48(2), 369–375 (2007) [Siberian Math. J. 48 (2), 295–299 (2007)]. · Zbl 1153.20017 [2] M. Aschbacher, ”Near subgroups of finite groups,” J. Group Theory 1(2), 113–129 (1998). · Zbl 0902.20010 [3] A. L. Myl’nikov, ”Minimal nongroup twisted subsets with involution,” Algebra i Logika 46(4), 459–482 (2007) [Algebra Logic 46 (4), 250–262 (2007)]. · Zbl 1155.20024 [4] A. L. Myl’nikov, ”Nilpotency of the derived subgroup of a finite twisted group,” Sibirsk. Mat. Zh. 47(5), 1117–1127 (2006) [Siberian Math. J. 47 (5), 915–923 (2006)]. · Zbl 1139.20020 [5] A. L. Myl’nikov, ”Finite minimal nontwisted groups,” Vestn. Krasnoyarsk Gos. Univ. 1, 71–76 (2005). [6] A. L. Myl’nikov, ”On finite minimal nontwisted groups,” Vestn. Krasnoyarsk Gos. Univ. 4, 164–169 (2005). [7] V. V. Belyaev and A. L. Myl’nikov, ”Estimation of the order of a group generated by a finite twisted subset,” Sibirsk. Mat. Zh. 49(6), 1235–1237 (2008) [Siberian Math. J. 49 (6), 985–987 (2008)]. · Zbl 1205.20028 [8] D. V. Veprintsev and A. L. Myl’nikov, ”Involutory decomposition of a group, and twisted subsets with few involutions,” Sibirsk. Mat. Zh. [Siberian Math. J.] 49(2), 274–279 (2008) [Siberian Math. J. 49 (2), 218–221 (2008)]. [9] A. L. Myl’nikov, ”Abelian twisted groups,” in Matem. Sistemy (Krasnoyarsk Gos. Agrar. Univ., Krasnoyrsk, 2005), Vol. 3, pp. 59–61 [in Russian]. [10] D. Gorenstein, Finite Groups (Harper & Row Publ., New York, 1968). [11] W. Feit and J. G. Thompson, ”Solvability of groups of odd order,” Pacific J. Math. 13(3), 775–1029 (1963). · Zbl 0124.26402
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.