Dominions in solvable groups

A. I. Budkin

Algebra i logika, Tome 54 (2015) no. 5, pp. 575-588.

Résumé

The dominion of a subgroup $H$ of a group $G$ in a class $M$ is the set of all elements $a\in G$ whose images are equal for all pairs of homomorphisms from $G$ to each group in $M$ that coincide on $H$. A group $H$ is absolutely closed in a class $M$ if, for any group $G$ in $M$ and any inclusion $H\le G$, the dominion of $H$ in $G$ (with respect to $M$) coincides with $H$ (i.e., $H$ is closed in $G$). We prove that every torsion-free nontrivial Abelian group is not absolutely closed in $\mathcal{AN}_c$. It is shown that if a subgroup $H$ of $G$ in $\mathcal N_c\mathcal A$ has trivial intersection with the commutator subgroup $G'$, then the dominion of $H$ in $G$ (with respect to $\mathcal N_c\mathcal A$) coincides with $H$. It is stated that the study of closed subgroups reduces to treating dominions of finitely generated subgroups of finitely generated groups.

Détail

Keywords: quasivariety, nilpotent group, extension of Abelian group by nilpotent group, dominion, closed subgroup.

Bibliographie

[1] A. I. Budkin, “Reshëtki dominionov universalnykh algebr”, Algebra i logika, 46:1 (2007), 26–45 | MR | Zbl

[2] A. I. Budkin, “Dominions in quasivarieties of universal algebras”, Stud. Log., 78:1/2 (2004), 107–127 | DOI | MR | Zbl

[3] S. A. Shakhova, “O reshëtkakh dominionov v kvazimnogoobraziyakh abelevykh grupp”, Algebra i logika, 44:2 (2005), 238–251 | MR | Zbl

[4] S. A. Shakhova, “Usloviya distributivnosti reshëtok dominionov v kvazimnogoobraziyakh abelevykh grupp”, Algebra i logika, 45:4 (2006), 484–499 | MR | Zbl

[5] S. A. Shakhova, “Ob odnom svoistve operatsii peresecheniya v reshetkakh dominionov kvazimnogoobrazii abelevykh grupp”, Izvestiya Alt. gos. un-ta, 2010, no. 1-1(65), 41–43

[6] S. A. Shakhova, “O suschestvovanii reshetki dominionov v kvazimnogoobraziyakh abelevykh grupp”, Izvestiya Alt. gos. un-ta, 2011, no. 1-1(69), 31–33

[7] A. I. Budkin, “Dominiony universalnykh algebr i proektivnye svoistva”, Algebra i logika, 47:5 (2008), 541–557 | MR | Zbl

[8] A. Magidin, “Dominions in varieties of nilpotent groups”, Commun. Algebra, 28:3 (2000), 1241–1270 | DOI | MR | Zbl

[9] A. Magidin, “Absolutely closed nil-2 groups”, Algebra Univers., 42:1/2 (1999), 61–77 | DOI | MR | Zbl

[10] A. I. Budkin, “O dominionakh v kvazimnogoobraziyakh metabelevykh grupp”, Sib. matem. zh., 51:3 (2010), 498–505 | MR | Zbl

[11] A. I. Budkin, “O dominione polnoi podgruppy metabelevoi gruppy”, Izvestiya Alt. gos. un-ta, 2010, no. 1-2(65), 15–19

[12] A. I. Budkin, “O dominionakh abelevykh podgrupp metabelevykh grupp”, Algebra i logika, 51:5 (2012), 608–622 | MR | Zbl

[13] A. I. Budkin, “Ob absolyutnoi zamknutosti abelevykh grupp bez krucheniya v klasse metabelevykh grupp”, Algebra i logika, 53:1 (2014), 15–25 | MR | Zbl

[14] A. I. Budkin, “O zamknutosti lokalno tsiklicheskoi podgruppy v metabelevoi gruppe”, Sib. matem. zh., 55:6 (2014), 1240–1249 | MR | Zbl

[15] A. I. Budkin, Kvazimnogoobraziya grupp, Izd-vo Alt. gos. un-ta, Barnaul, 2002

[16] A. I. Budkin, V. A. Gorbunov, “K teorii kvazimnogoobrazii algebraicheskikh sistem”, Algebra i logika, 14:2 (1975), 123–142 | MR | Zbl

[17] V. A. Gorbunov,, Algebraicheskaya teoriya kvazimnogoobrazii, Sibirskaya shkola algebry i logiki, Nauch. kniga, Novosibirsk, 1999

[18] A. I. Maltsev, Algebraicheskie sistemy, Nauka, M., 1970 | MR

[19] M. I. Kargapolov, Yu. I. Merzlyakov, Osnovy teorii grupp, Nauka, M., 1982 | MR

[20] G. Baumslag, “Wreath products and $p$-groups”, Proc. Camb. Philos. Soc., 55:3 (1959), 224–231 | DOI | MR | Zbl

[21] P. Hall, “Finiteness conditions for soluble groups”, Proc. Lond. Math. Soc. III Ser., 4:16 (1954), 419–436 | DOI | MR | Zbl