Document Zbl 1436.20042

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Lond. Math. Soc.</a> <a href="/?q=in%3A417785" title="Articles in this Issue">50, No. 5, 832-844 (2018)</a>. </div> <div class="abstract">Summary: It is well known that a nontrivial commutator in a free group is never a proper power. We prove a theorem that generalizes this fact and has several worthwhile corollaries. For example, an equation $[x_1,y_1]\dots[x_k,y_k]=z^n$, where $n\geqslant2k$, in the free product $\mathcal{F}$ of groups without nontrivial elements of order $\leqslant n$ implies that $z$ is conjugate to an element of a free factor of $\mathcal{F}$. If a nontrivial commutator in a free group factors into a product of elements which are conjugate to each other then all these elements are distinct.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6968898">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A6968898">Cited in <strong>7</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A20E06" title="MSC2020">20E06</a> </td> <td class="space"> Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20F06" title="MSC2020">20F06</a> </td> <td class="space"> Cancellation theory of groups; application of van Kampen diagrams </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20F12" title="MSC2020">20F12</a> </td> <td class="space"> Commutator calculus </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20F70" title="MSC2020">20F70</a> </td> <td class="space"> Algebraic geometry over groups; equations over groups </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A57M07" title="MSC2020">57M07</a> </td> <td class="space"> Topological methods in group theory </td> </tr></table> </div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1436.20042" data-ciurl="/ci/06968898" data-biburl="/bibtex/06968898.bib" data-amsurl="/amsrefs/06968898.bib" data-xmlurl="/xml/06968898.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06968898.pdf" title="Zbl 1436.20042 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1112/blms.12188" aria-label="DOI for “Quasiperiodic and mixed commutator factorizations in free products of groups”" title="10.1112/blms.12188">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1702.01379"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Baumslag, G.; Steinberg, A., Residual nilpotence and relations in free groups, Bull. 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