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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Residual solvability of an equation in nilpotent groups


Author: Peter F. Stebe
Journal: Proc. Amer. Math. Soc. 54 (1976), 57-58
MathSciNet review: 0387410
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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$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0387410-2
PII: S 0002-9939(1976)0387410-2
Keywords: Residual property, nilpotent group
Article copyright: © Copyright 1976 American Mathematical Society