SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Length function</title>

</head>

<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Length function</span></span>

</h1>

</header>

<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In the <a href="Mathematics" title="Mathematics">mathematical</a> field of <a href="Geometric_group_theory" title="Geometric group theory">geometric group theory</a>, a <b>length function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that assigns a number to each element of a <a href="Group_(mathematics)" title="Group (mathematics)">group</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>A <b>length function</b> <i>L</i> : <i>G</i> → <b>R</b><sup>+</sup> on a <a href="Group_(mathematics)" title="Group (mathematics)">group</a> <i>G</i> is a function satisfying:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msup>

</mrow>

</msup>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}L(e)&=0,\\L(g^{-1})&=L(g)\\L(g_{1}g_{2})&\leq L(g_{1})+L(g_{2}),\quad \forall g_{1},g_{2}\in G.\end{aligned}}}</annotation>

</semantics>

</math></span><img src="./b05ebe0b10654813e4e733a0bf5de8f4c7cbeda6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:40.844ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}L(e)&=0,\\L(g^{-1})&=L(g)\\L(g_{1}g_{2})&\leq L(g_{1})+L(g_{2}),\quad \forall g_{1},g_{2}\in G.\end{aligned}}}" loading="lazy"></span></dd></dl>

<p>Compare with the <a href="Axiom" title="Axiom">axioms</a> for a <a href="Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> and a <a href="Filtered_algebra" title="Filtered algebra">filtered algebra</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Word_metric">Word metric</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Word_metric" title="Word metric">Word metric</a></div>

<p>An important example of a length is the <a href="Word_metric" title="Word metric">word metric</a>: given a <a href="Presentation_of_a_group" title="Presentation of a group">presentation of a group</a> by generators and relations, the length of an element is the length of the shortest word expressing it.

</p><p><a href="Coxeter_group" title="Coxeter group">Coxeter groups</a> (including the <a href="Symmetric_group" title="Symmetric group">symmetric group</a>) have combinatorially important length functions, using the simple reflections as generators (thus each simple reflection has length 1). See also: <a href="Length_of_a_Weyl_group_element" title="Length of a Weyl group element">length of a Weyl group element</a>.

</p><p>A <a href="Longest_element_of_a_Coxeter_group" title="Longest element of a Coxeter group">longest element of a Coxeter group</a> is both important and unique up to conjugation (up to different choice of simple reflections).

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>A group with a length function does <i>not</i> form a <a href="Filtered_group" class="mw-redirect" title="Filtered group">filtered group</a>, meaning that the <a href="Sublevel_set" class="mw-redirect" title="Sublevel set">sublevel sets</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{i}:=\{g\mid L(g)\leq i\}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S_{i}:=\{g\mid L(g)\leq i\}}</annotation>

</semantics>

</math></span><img src="./179dd807c3038231018f7dd0b7d4a2d85d248f27.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.757ex; height:2.843ex;" alt="{\displaystyle S_{i}:=\{g\mid L(g)\leq i\}}" loading="lazy"></span> do not form <a href="Subgroup" title="Subgroup">subgroups</a> in general.

</p><p>However, the <a href="Group_ring" title="Group ring">group algebra</a> of a group with a length functions forms a <a href="Filtered_algebra" title="Filtered algebra">filtered algebra</a>: the axiom <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L(gh)\leq L(g)+L(h)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L(gh)\leq L(g)+L(h)}</annotation>

</semantics>

</math></span><img src="./a8c4ca950318b2c5c6acd17dfd5401cad16f4d2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.025ex; height:2.843ex;" alt="{\displaystyle L(gh)\leq L(g)+L(h)}" loading="lazy"></span> corresponds to the filtration axiom.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist">

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFLyndon1963" class="citation cs2">Lyndon, Roger C. (1963), "Length functions in groups", <i>Mathematica Scandinavica</i>, <b>12</b>: <span class="nowrap">209–</span>234, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.7146%2Fmath.scand.a-10684">10.7146/math.scand.a-10684</a>, <a href="JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/24489388">24489388</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0163947">0163947</a></cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFHarrison1972" class="citation cs2">Harrison, Nancy (1972), "Real length functions in groups", <i>Transactions of the American Mathematical Society</i>, <b>174</b>: <span class="nowrap">77–</span>106, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F1996098">10.2307/1996098</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0308283">0308283</a></cite></span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFChiswell1976" class="citation cs2">Chiswell, I. M. (1976), "Abstract length functions in groups", <i>Mathematical Proceedings of the Cambridge Philosophical Society</i>, <b>80</b> (3): <span class="nowrap">451–</span>463, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100053093">10.1017/S0305004100053093</a>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0427480">0427480</a></cite></span>

</li>

</ol></div></div>

<p><i>This article incorporates material from Length function on <a href="PlanetMath" title="PlanetMath">PlanetMath</a>, which is licensed under the Creative Commons Attribution/Share-Alike License.</i>

</p></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-01-23" href="https://en.wikipedia.org/wiki/?title=Length_function&oldid=1271346194">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>