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The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. <a href="https://mathscinet.ams.org/mathscinet/search/journal/profile?groupId=3285" target="new">Subscribers to MathSciNet</a> may click through for more detailed information. </div> </p> </section> </div> </header> <main> <section class="row"> <nav class="col-md-3">  <nav class="journals-list-group"> <script type="text/javascript" src="/journals/javascript/tran_nav.js"></script> <a href="/journals/journals"><i class="fa-sharp fa-solid fa-house"></i> Journals Home</a> <a href="/epubsearch">eContent Search</a> <a href="/abouttran" class="">About Transactions of the American Mathematical Society</a> <a href="/tranedit" class="">Editorial Board</a> <a href="/tran/transubmit.html" class="">Author and Submission Information</a> <a href="/policies/policies" class="" target="_blank">Journal Policies</a> <a href="/subscriberinfo" class="" target="_blank">Subscription Information</a> <a href="/jemail" class="" target="_blank"> Journal Email Notifications</a> <hr> <a href="/pairing/pair_my_device.html"><i class="fa fa-globe" aria-hidden="true"></i> Remote Access</a> </nav> </nav> <article class="col-md-9"> <section>   <div class="row">  <ul class="nav nav-tabs"> <li role="presentation" class=""> <script type="text/javascript" src="/journals/javascript/tran_nav.js"></script> <a href="javascript: goto_most_recent_tran()" aria-controls="currissue" role="tab"> Current volume </a> </li> <li role="presentation" class="">  <a href="/cgi-bin/mstrack/accepted_papers/tran?active=press" aria-controls="press" role="tab">Articles in press </a> </li> <li role="presentation" class=""> <a href="/journals/tran/0000-000-00?active=recent"> Recently published</a> </li> <li role="presentation" class=""> <a href="/journals/tran/all_issues.html?active=allissues"> All issues : 1900 – Present </a> </li> </ul> <script src="/js/abstracts/ajaxcall.js"></script> </div> <p> </p> <style> #navCell { display: none;} </style> <script src="/journals/javascript/references.js" type="text/javascript"></script> <script async src="/js/mj.js"></script>  <script src="https://platform-api.sharethis.com/js/sharethis.js#property=598c93d1e2587a001253a011&product=unknown" async="async"></script>  <div class="sharethis-inline-share-buttons"></div>   <article class="col-md-12"> <header> <h2>The solution of length four equations over groups<br> <small>HTML articles powered by AMS MathViewer <a href="/AMSMathViewer"><i class="fa fa-info-circle" aria-label="about MathViewer"></i></a></small></h2> <dl> <dd>by <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Edjvet%2C%20Martin">Martin Edjvet</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Howie%2C%20James">James Howie</a>  </dd> <dd>Trans. Amer. Math. Soc. <strong>326</strong> (1991), 345-369 </dd> <dd> DOI: https://doi.org/10.1090/S0002-9947-1991-1002920-5 </dd> <dd> </dd> <dd class="mathviewerPDFLink"> <a href="S0002-9947-1991-1002920-5.pdf"><small>PDF</small></a> | <a href="https://www.copyright.com/openurl.do?issn=1088-6850&WT.mc.it=American%20Mathematical%20Society" target="_blank">Request permission</a> </dd> </dl> </header> <section id="Abstract"> <h2>Abstract:</h2> Let $G$ be a group, $F$ the free group generated by $t$ and let $r(t) \in G \ast F$. The equation $r(t) = 1$ is said to have a solution over $G$ if it has a solution in some group that contains $G$. This is equivalent to saying that the natural map $G \to \langle G \ast F|r(t)\rangle$ is injective. There is a conjecture (attributed to M. Kervaire and F. Laudenbach) that injectivity fails only if the exponent sum of $t$ in $r(t)$ is zero. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of $t$ in $r(t)$ is equal to four. </section> <section id="references"> <details> <summary>References</summary> <ul><li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Collins%2C%20D.%20J.">D. J. Collins</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Huebschmann%2C%20J.">J. Huebschmann</a>, <em>Spherical diagrams and identities among relations</em>, Math. Ann. <strong>261</strong> (1982), no. 2, 155–183. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=675732">MR <strong>675732</strong></a>, DOI <a href="https://doi.org/10.1007/BF01456216">10.1007/BF01456216</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Edjvet%2C%20M.">M. Edjvet</a>, <em>Solutions of certain sets of equations over groups</em>, Groups—St. Andrews 1989, Vol. 1, London Math. Soc. Lecture Note Ser., vol. 159, Cambridge Univ. Press, Cambridge, 1991, pp. 105–123. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1123977">MR <strong>1123977</strong></a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Gerstenhaber%2C%20Murray">Murray Gerstenhaber</a> and <a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Rothaus%2C%20Oscar%20S.">Oscar S. Rothaus</a>, <em>The solution of sets of equations in groups</em>, Proc. Nat. Acad. Sci. U.S.A. <strong>48</strong> (1962), 1531–1533. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=166296">MR <strong>166296</strong></a>, DOI <a href="https://doi.org/10.1073/pnas.48.9.1531">10.1073/pnas.48.9.1531</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Howie%2C%20James">James Howie</a>, <em>The solution of length three equations over groups</em>, Proc. Edinburgh Math. Soc. (2) <strong>26</strong> (1983), no. 1, 89–96. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=695646">MR <strong>695646</strong></a>, DOI <a href="https://doi.org/10.1017/S0013091500028108">10.1017/S0013091500028108</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Levin%2C%20Frank">Frank Levin</a>, <a href="/bull/1962-68-06/S0002-9904-1962-10868-4/" class="amsJournalReferenceTitle"><em>Solutions of equations over groups</em></a>, <a href="/bull/">Bull. Amer. Math. Soc.</a> <a href="/bull/1962-68-06/"><strong>68</strong></a> (1962), 603–604. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=142643">MR <strong>142643</strong></a>, DOI <a href="/bull/1962-68-06/S0002-9904-1962-10868-4/">10.1090/S0002-9904-1962-10868-4</a></li> <li><a href="https://mathscinet.ams.org/mathscinet/search/authors.html?authorName=Lyndon%2C%20Roger%20C.">Roger C. Lyndon</a>, <em>Equations in groups</em>, Bol. Soc. Brasil. 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(2) <strong>106</strong> (1977), no. 3, 599–612. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=467760">MR <strong>467760</strong></a>, DOI <a href="https://doi.org/10.2307/1971070">10.2307/1971070</a></li> </ul> </details> </section> <section id="similar"> <details> <summary>Similar Articles</summary> <ul> <li> Retrieve articles in <em>Transactions of the American Mathematical Society</em> with MSC: <a href="/joursearch/servlet/DoSearch?f1=msc&v1=20E06&jrnl=one&onejrnl=tran">20E06</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=20F05&jrnl=one&onejrnl=tran">20F05</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=20F06&jrnl=one&onejrnl=tran">20F06</a> </li> <li> Retrieve articles in all journals with MSC: <a href="/joursearch/servlet/DoSearch?f1=msc&v1=20E06">20E06</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=20F05">20F05</a>, <a href="/joursearch/servlet/DoSearch?f1=msc&v1=20F06">20F06</a> </ul> </details> </section> <section id="additionalinformation"> <details open> <summary>Bibliographic Information</summary> <ul> <li>© Copyright 1991 American Mathematical Society</li> <li>Journal: Trans. 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Soc. <strong>326</strong> (1991), 345-369 </li> <li>MSC: Primary 20E06; Secondary 20F05, 20F06 <li>DOI: https://doi.org/10.1090/S0002-9947-1991-1002920-5</li> <li>MathSciNet review: <a href = "https://mathscinet.ams.org/mathscinet-getitem?mr=1002920">1002920</a></li> </ul> </details> </section> </article> </section> </article> </section> </main> </div>   <hr class="mx-auto"> <footer id="pagefooter" class="text-center"> <div class="container text-center"> <div class="row">  <div class="col-12 mb-1"> <p> American Mathematical Society · <a href="https://goo.gl/maps/ZvJdkQEC79p" title="Google map 201 Charles Street Providence, Rhode Island" aria-label="Google map to AMS headquarters"> 201 Charles Street Providence, Rhode Island 02904-2213 </a> · <a href="/reach" aria-label="Contact Us">Contact Us</a> </p> </div>  <script src="https://kit.fontawesome.com/2ad6d7fbd6.js" crossorigin="anonymous" defer></script>  <nav class="col-12 mb-1" aria-label="Social media links"> <ul class="list-inline"> <li class="list-inline-item"> <a href="https://www.facebook.com/amermathsoc" class="link-dark" title="Follow us on Facebook" aria-label="View the AMS on Facebook"> <i class="fa-brands fa-facebook-f" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://bsky.app/profile/amermathsoc.bsky.social" class="link-dark" title="Follow us on Bluesky" aria-label="View the AMS on Bluesky"> <i class="fa-brands fa-bluesky"></i></a></li> <li class="list-inline-item"> <a href="https://twitter.com/amermathsoc" class="link-dark" title="Follow us on X" aria-label="View the AMS on X"> <i class="fa-brands fa-x-twitter" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.linkedin.com/company/51001" class="link-dark" title="Follow us on LinkedIn" aria-label="View the AMS on LinkedIn"> <i class="fa-brands fa-linkedin-in" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.instagram.com/amermathsoc" class="link-dark" title="Follow us on Instagram" aria-label="View the AMS on Instagram"> <i class="fa-brands fa-instagram" aria-hidden="true"></i></a></li> <li class="list-inline-item"> <a href="https://www.youtube.com/amermathsoc" class="link-dark" title="Follow us on YouTube" aria-label="View the AMS on YouTube"> <i class="fa-brands fa-youtube" aria-hidden="true"></i></a></li> </ul> </nav>  <div class="col-12"> <p><small> © <a href="/copyright" aria-label="Copyright policy">Copyright</a> <span id="copyYear">2024</span> American Mathematical Society · <a href="/privacy" aria-label="Privacy statement">View Our Privacy Statement</a> · <a href="/tou" aria-label="Terms of use">Terms of Use</a> · <a href="/accessibility" aria-label="Accessibility policy">Accessibility and AMS Online Content</a> </small></p> <p><small> AMS, American Mathematical Society, the tri-colored AMS logo, and Advancing research, Creating connections, are trademarks and service marks of the American Mathematical Society and registered in the U.S. Patent and Trademark Office. </small></p> </div> </div> </div> <script> // Dynamically update the copyright year document.getElementById('copyYear').textContent = new Date().getFullYear(); </script> </footer>