Illinois Journal of Mathematics

Embedding of groups and quadratic equations over groups

D. F. Cummins and S. V. Ivanov

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We prove that, for every integer $n\ge2$, a finite or infinite countable group $G$ can be embedded into a 2-generated group $H$ in such a way that the solvability of quadratic equations of length at most $n$ is preserved, that is, every quadratic equation over $G$ of length at most $n$ has a solution in $G$ if and only if this equation, considered as an equation over $H$, has a solution in $H$.

Article information

Illinois J. Math., Volume 60, Number 1 (2016), 99-115.

Received: 30 August 2015
Revised: 19 July 2016
First available in Project Euclid: 21 June 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F05: Generators, relations, and presentations 20F06: Cancellation theory; application of van Kampen diagrams [See also 57M05] 20F70: Algebraic geometry over groups; equations over groups


Cummins, D. F.; Ivanov, S. V. Embedding of groups and quadratic equations over groups. Illinois J. Math. 60 (2016), no. 1, 99--115. doi:10.1215/ijm/1498032025.

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