On the residual nilpotence of free products of nilpotent groups with central amalgamated subgroups

A. V. Rozov ; E. V. Sokolov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 6 (2016), pp. 34-44

Résumé

Let $G$ be a free product of nilpotent groups $A$ and $B$ with proper amalgamated subgroups $H$ and $K$. We state that if $H$ and $K$ lie in the centers of $A$ and $B$, respectively, then $G$ is residually nilpotent if and only if the ordinary free product of $A/H$ and $B/K$ possesses the same property. We also prove that if $\pi$ is a non-empty set of primes, $H$ is central in $A$, and $K$ is normal in $B$, then $G$ is residually $\pi$-finite nilpotent if and only if $G$ is residually $\pi$-finite and the free product of $A/H$ and $B/K$ is residually $\pi$-finite nilpotent. We obtain two corollaries of the second result for the cases when $A$ and $B$ have finite ranks or finite numbers of generators. In particular, we prove that if $A$ and $B$ are finitely generated, $H$ is central in $A$, and $K$ is normal in $B$, then $G$ is residually $\pi$-finite nilpotent if and only if the periodic parts of $A$ and $B$ are $\pi$-groups and the periodic parts of $A/H$ and $B/K$ are $p$-groups for some prime $p$ which belongs to $\pi$.

Détail

Keywords: nilpotent group, generalized free product of groups, residual nilpotence, residual finite nilpotence.

Bibliographie

[1] Magnus W., “Beziehungen zwischen Gruppen und idealen in einem speziellen Ring”, Math. Ann., 111 (1935), 259–280 | DOI | MR

[2] Mal'cev A. I., “Generalized nilpotent algebras and their adjoint groups”, American Mathematical Society Translations: Series 2, 69 (1968), 1–21 | DOI | Zbl

[3] Lichtman A. I., “Necessary and sufficient conditions for the residual nilpotence of free products of groups”, J. Pure Appl. Algebra, 12 (1978), 49–64 | DOI | MR | Zbl

[4] Raptis E., Varsos D., “The residual nilpotence of HNN-extensions with base group a finite or a f.g. abelian group”, J. Pure Appl. Algebra, 76:2 (1991), 167–178 | DOI | MR | Zbl

[5] Azarov D. N., “On the residual nilpotence of free products of free groups with cyclic amalgamation”, Mathematical Notes, 64:1 (1998), 3–7 | DOI | DOI | MR | Zbl

[6] Azarov D. N., Ivanova E. A., “To the question on the residual nilpotence of free product with amalgamation of locally nilpotent groups”, Nauchnye Trudy Ivanovskogo Gosudarstvennogo Universiteta. Matematika — Scientific Works of Ivanovo State University. Mathematics, 1999, no. 2, 5–7

[7] Ivanova E. A., “On the residual nilpotence of a free product with an amalgamated subgroup of two Abelian groups by nilpotent groups”, Chebyshevskii Sbornik, 3:1 (2002), 72–77 | Zbl

[8] Ivanova E. A., On the residual nilpotence of generalized free products of groups, Abstract of Phys. and Math. Cand. Diss., Ivanovo, 2004

[9] Ivanova E. A., “The residual nilpotence of free product of two groups with finite amalgamated subgroups”, Vestnik Ivanovskogo gosudarstvennogo universiteta — Ivanovo State University Bulletin. Ser.: Biology, Chemistry, Physics, Mathematics, 3 (2004), 120–125 | Zbl

[10] Azarov D. N., Ivanova E. A., “The residual properties of free products of finitely generated nilpotent groups with cyclic amalgamation”, Vestnik Ivanovskogo gosudarstvennogo universiteta — Ivanovo State University Bulletin. Ser.: Biology, Chemistry, Physics, Mathematics, 2008, no. 3, 56–62

[11] Savelicheva N. S., Sokolov E. V., “A necessary condition of the residual nilpotence of an HNN-extension of a nilpotent group”, Vestnik Ivanovskogo gosudarstvennogo universiteta — Ivanovo State University Bulletin. Ser.: Natural, Social Sciences, 2015, no. 2, 64–68

[12] Varsos D., “The residual nilpotence of the fundamental group of certain graphs of groups”, Houston J. Math., 22:2 (1996), 233–248 | MR | Zbl

[13] Gruenberg K. W., “Residual properties of infinite soluble groups”, Proc. London Math. Soc. Ser. 3, 7 (1957), 29–62 | DOI | MR | Zbl

[14] Sokolov E. V., “On the application of D. I. Moldavanskii's method to the study of the approximability of HNN-extensions by root classes of groups”, Vestnik Ivanovskogo gosudarstvennogo universiteta — Ivanovo State University Bulletin. Ser.: Natural, Social Sciences, 2016, no. 2, 87–103

[15] Rozov A. V., “On the residual $\pi$-finiteness of some free products of groups with central amalgamated subgroups”, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika - Tomsk State University Journal of Mathematics and Mechanics, 2016, no. 2(40), 37–44 | DOI

[16] Rozov A. V., “On nilpotent groups of finite rank”, Matematika i ee prilozheniya — Mathematics and Its Applications, 2012, no. 9, 41–44

[17] Magnus W., Karrass A., Solitar D., Combinatorial group theory: presentations of groups in terms of generators and relations, Interscience Publishers, New York–London–Sydney, 1966 | MR | MR | Zbl

[18] Baumslag G., “On the residual finiteness of generalized free products of nilpotent groups”, Trans. Amer. Math. Soc., 106 (1963), 193–209 | DOI | MR | Zbl

[19] Tumanova E. A., “On the root-class residuality of generalized free products with a normal amalgamation”, Russian Mathematics, 59:10 (2015), 23–37 | DOI | MR | Zbl

[20] Moldavanskii D. I., “On the intersection of finite index subgroups of certain generalized free products of groups”, Vestnik Ivanovskogo gosudarstvennogo universiteta - Ivanovo State University Bulletin. Ser.: Biology, Chemistry, Physics, Mathematics, 2008, no. 3, 114–122

[21] Kargapolov M. I., Merslyakov Yu. I., Foundations of the Theory of Groups, Nauka, M., 1977 | MR