# A.V. Martsinkevich, N.T. Vorob’ev. Products and joins of locally normal Fitting class ... С. 152-157

Let $\pi$ be a nonempty set of primes. A nontrivial Fitting class $\mathfrak{F}$ is said to be normal in the class $\mathfrak{S}_\pi$ of all finite soluble $\pi$-groups or $\pi$-normal (we write $\mathfrak{F\trianglelefteq S}_\pi$) if $\mathfrak{F\subseteq S}_\pi$ and the $\mathfrak{F}$-radical of every $\pi$-group $G$ is a $\mathfrak{F}$-maximal subgroup of $G$. If $\pi$ is the set of all primes, then $\mathfrak{F}$ is called normal. The product $\mathfrak{FH}$ of Fitting classes $\mathfrak{F}$ and $\mathfrak{H}$ is called $\pi$-normal if $\mathfrak{FH}$ is a $\pi$-normal Fitting class. We prove the existence of $\pi$-normal products of Fitting classes factorizable by non-$\pi$-normal factors. Assume that $\mathbb{P}$ is the set of all primes, $\varnothing\neq\pi\subseteq\mathbb{P}$, $\mathfrak{F}$ is some Fitting class of $\pi$-groups, and $\omega=\sigma(\mathfrak{F})$ is the set of all prime divisors of all groups from $\mathfrak{F}$. It is proved that if $\mathfrak{F^2=F}$ and $\mathfrak{H}$ is the class of all $\pi$-groups with central $\omega$-socle, then the product $\mathfrak{FH}$ is $\pi$-normal although each of the factors $\mathfrak{F}$ and $\mathfrak{H}$ is not $\pi$-normal. The lattice join $\mathfrak{F\vee H}$ of Fitting classes $\mathfrak{F}$ and $\mathfrak{H}$ is the Fitting class generated by $\mathfrak{F\cup H}$. If $\mathfrak{F\vee H}$ is a $\pi$-normal Fitting class, then $\mathfrak{F\vee H}$ is called $\pi$-normal. Let $\mathfrak{F}$ and $\mathfrak{H}$ be Fitting classes of $\pi$-groups. We prove that the lattice join $\mathfrak{F\vee H}$ is a $\pi$-normal Fitting class if and only if $\mathfrak{F}$ or $\mathfrak{H}$ is a $\pi$-normal Fitting class.

Keywords: $\mathfrak{F}$-radical, Fitting class, $\pi$-normal Fitting class, join of Fitting classes.

The paper was received by the Editorial Office on November 11, 2017.

Funding Agency:

1) Belarusian Republican Foundation for Fundamental Research  (Grant Number Ф17М-064);

2) National Academy of Sciences of Belarus, Ministry of Education of the Republic of Belarus.

Anna Veslavovna Martsinkevich. doctoral student, Masherov Vitebsk State University, Vitebsk, 210038 Belarus, e-mail: hanna-t@mail.ru.

Nikolai Timofeevich Vorob’ev. Dr. Phys.-Math. Sci., Prof., Masherov Vitebsk State University, Vitebsk, 210038 Belarus, e-mail: vorobyovnt@tut.by.

REFERENCES

1.   Doerk K., Hawkes T. Finite soluble groups. Berlin; New York: Walter de Gruyter & Co, 1992, Ser.: De Gruyter Expo. Math., 4, 891 p. ISBN: 978-3-11-087013-8 .

2.   Blessenohl D., Gasch$\ddot{\mathrm{u}}$tz W.  $\ddot{\mathrm{U}}$ber normale Schunk- und Fittingklassen. Math. Z., 1970, vol. 118, no. 1, pp. 1–8. doi: 10.1007/BF01109888 .

3.   Vorob’ev N.T., Martsinkevich A.V. Finite $\pi$-groups with normal injectors. Sib. Math. J., 2015, vol. 56, no. 4, pp. 624–630. doi: 10.17377/smzh.2015.56.406 .

4.   Cossey J. Products of Fitting classes. Math. Z., 1975, vol. 141, no. 3, pp. 289–295. doi: 10.1007/BF01247314 .

5.   Kourovka notebook: Unsolved problems of group theory. 11th ed., IM RAN, Novosibirsk, 1990, 126 p. (in Russian).

6.   Vorob’ev N.T. On the factorization of local and non-local products of finite groups of non-local formations. Proc. 7th Reg. Sci. Sess. Math., Sect. Algebra and Number Theory. Kalsk, 1990, pp. 9–13.

7.   Vedernikov V.A. Local formations of finite groups. Math. Notes, 1989, vol. 46, no. 6, pp. 910–913. doi: 10.1007/BF01158624 .

8.   Vorob’ev N.T., Skiba A.N. Local products of non-local Fitting classses. Voprosy algebry, 1995, no. 8, pp. 55–58 (in Russian).

9.   Shpakov V.V., Vorobyev N.T. Local factorisations of nonlocal Fitting classes. Discrete Math. Appl., 2008, vol. 18, no. 4, pp. 439–446. doi: 10.1515/DMA.2008.032.

10.   Lockett F.P. The Fitting class $\mathfrak{F^*}$. Math. Z., 1974, vol. 137, no. 2, pp. 131–136. doi: 10.1007/BF01214854 .

11.   Cusack E. The join of two Fitting classes. Math. Z., 1979, vol. 167, no. 1, pp. 37–47. doi: 10.1007/BF01215242 .

12.   Beidleman J.C. On products and normal Fitting classes. Arch. Math., 1977, vol. 28, no. 1, pp. 347–356. doi: 10.1007/BF01223934 .

13.   GaschЈutz W. Lectures of subgroups of Sylow type in finite soluble groups. Notes on pure mathematics, 1979, no. 11, pp. 1–100.

14.   Savelyeva N.V., Vorob’ev N.T. Maximal on strong π-containment Fitting classes. Izv. Gomel. Gos. Univ. im. F. Skoriny, 2008, no. 2(47), pp. 157–168 (in Russian).