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Register a new userOn groups presented by inverse-closed finite convergent length-reducing rewriting systems
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Added by
Murray Elder
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate triangles are uniformly bounded. This leads to a new algebraic result: the group is plain (isomorphic to the free product of finitely many finite groups and copies of $mathbb Z$) if and only if a certain relation on the set of non-trivial finite-order elements of the group is transitive on a bounded set. We use this to prove that deciding if a group presented by an inverse-closed finite convergent length-reducing rewriting system is not plain is in $mathsf{NP}$. A yes answer would disprove a longstanding conjecture of Madlener and Otto from 1987. We also prove that the isomorphism problem for plain groups presented by inverse-closed finite convergent length-reducing rewriting systems is in $mathsf{PSPACE}$.
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An abstract group $G$ is called totally 2-closed if $H = H^{(2),Omega}$ for any set $Omega$ with $Gcong Hleqtextrm{Sym}_Omega$, where $H^{(2),Omega}$ is the largest subgroup of symmetric group of $Omega$ whose orbits on $OmegatimesOmega$ are the same orbits of $H$. In this paper, we prove that the Fitting subgroup of a totally 2-closed group is a totally 2-closed group. We also conjecture that a finite group $G$ is totally 2-closed if and only if it is cyclic or a direct product of a cyclic group of odd order with a generalized quaternion group. We prove the conjecture in the soluble case, and reduce the general case to groups $G$ of shape $Zcdot X$, with $Z = Z(G)$ cyclic, and $X$ is a finite group with a unique minimal normal subgroup, which is nonabelian
We prove that a group is presented by finite convergent length-reducing rewriting systems where each rule has left-hand side of length 3 if and only if the group is plain. Our proof goes via a new result concerning properties of embedded circuits in geodetic graphs, which may be of independent interest in graph theory.
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