V.S. Monakhov. A metanilpotency criterion for a finite solvable group ... P.253-256

Denote by $|x|$ the order of an element $x$ of a group. An element of a group is called primary if its order is a nonnegative integer power of a prime. If $a$ and $b$ are primary elements of coprime orders of a group, then the commutator $a^{-1}b^{-1}ab$ is called a $\star$-commutator. The intersection of all normal subgroups of a group such that the quotient groups by them are nilpotent is called the nilpotent residual of the group. It is established that the nilpotent residual of a finite group is generated by commutators of primary elements of coprime orders. It is proved that the nilpotent residual of a finite solvable group is nilpotent if and only if $|ab|\ge|a||b|$ for any $\star$-commutators of $a$ and $b$ of coprime orders.

Keywords: finite group, formation, residual, nilpotent group, commutator.

The paper was received by the Editorial Office on August 30, 2017

Viktor Stepanovich Monakhov, Dr. Phys.-Math. Sci., Prof., Francisk Skorina Gomel State University,
Gomel, 246019, Republic of Belarus, e-mail: victor.monakhov@gmail.com.

REFERENCES

1.   Huppert B. Endliche Gruppen I. Berlin etc.: Springer, 1967. 793 p. doi: 10.1007/978-3-642-64981-3.

2.   Bastos R.,  Shumyatsky P. A sufficient condition for nilpotency of the commutator subgroup. Siberian Math. J., 2016, vol. 57, no. 5, pp. 762-763. doi: 10.1134/S0037446616050037.

3.   Monakhov V.S. The nilpotency criterion for the derived subgroup of a finite group [e-resource]. 2017. Preprint available at https://arxiv.org/abs/1704.01746.

4.   Schmidt O.Yu.  Groups whose all subgroups are special. Mat.Sb., 1924, vol. 31, pp. 366-372.

5.   Monakhov V.S. Podgruppy Shmidta, ikh sushchestvovanie i nekotorye prilozheniya [The Schmidt subgroups, its existence, and some of their applications]. Tr. Ukrain. Mat. Congr., Kiev, 2002, Section 1, pp. 81-90. (In Russian)

6.   Tschunichin S. Uber spezielle Gruppen. Mat. Sb., 1929, vol. 36, no. 2, pp. 135-137 (in Russian).

7.   Doerk K., Hawkes T. Finite soluble groups. Berlin, N. Y.: Walter de Gruyter, 1992. 891 p. ISBN: 978-3-11-087013-8.

8.   Beidleman J., Heineken H. Minimal non-$\mathfrak F$-groups. Ricerche Mat., 2009, vol. 58, no. 1, pp. 33-41. doi: 10.1007/s11587-009-0044-2.