Residual solvability of an equation in nilpotent groups
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- by Peter F. Stebe PDF
- Proc. Amer. Math. Soc. 54 (1976), 57-58 Request permission
Abstract:
Let $G$ be a finitely generated nilpotent group. Let ${S_1}$ and ${S_2}$ be subgroups of $G$. Let ${S_1}{S_2}$ be the set of all products ${g_1}{g_2}$, where ${g_i}$ is an element of ${S_i}$. Let $g$ be an element of $G$. It is shown that either $g$ is an element of ${S_1}{S_2}$ or there is a normal subgroup $N$ of finite index in $G$ such that $gN$ does not meet ${S_1}{S_2}$. This result implies: (a) There is an algorithm to determine whether or not $g$ is an element of ${S_1}{S_2}$. (b) Given elements $a,b$ and $c$ of $G$, there is an algorithm to determine whether there exist integers $n$ and $m$ such that $a = {b^m}{c^n}$. (c) Finitely generated nilpotent groups are subgroup separable (a result of K. Toh). (d) Given elements $a$ and $b$ of $G$ and a subgroup $S$ of $G$, there is an algorithm to determine whether or not $a$ is an element of $SbS$.References
- Gilbert Baumslag, Lecture notes on nilpotent groups, Regional Conference Series in Mathematics, No. 2, American Mathematical Society, Providence, R.I., 1971. MR 0283082
- Peter F. Stebe, Conjugacy separability of certain free products with amalgamation, Trans. Amer. Math. Soc. 156 (1971), 119–129. MR 274597, DOI 10.1090/S0002-9947-1971-0274597-5 K. Toh, Problems concerning residual finiteness in nilpotent groups, Dept. of Math., Univ. of Malaya, Kuala Lumpur, Malaysia.
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 54 (1976), 57-58
- DOI: https://doi.org/10.1090/S0002-9939-1976-0387410-2
- MathSciNet review: 0387410