Conjugacy classes, characters and products of elements

Guralnick Robert M.; Moretó Alexander

摘要

Abstract > Recently, Baumslag and Wiegold proved that a finite group G is nilpotent if and only if <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201800403:mana201800403-math-0001" wiley:location="equation/mana201800403-math-0001.png"> <mat:mrow> <mat:mi>o</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>x</mat:mi> <mat:mi>y</mat:mi> <mat:mo stretchy="false">)</mat:mo> <mat:mo>=</mat:mo> <mat:mi>o</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>x</mat:mi> <mat:mo stretchy="false">)</mat:mo> <mat:mi>o</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>y</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> </mat:math> for every <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201800403:mana201800403-math-0002" wiley:location="equation/mana201800403-math-0002.png"> <mat:mrow> <mat:mi>x</mat:mi> <mat:mo>,</mat:mo> <mat:mi>y</mat:mi> <mat:mo>∈</mat:mo> <mat:mi>G</mat:mi> </mat:mrow> </mat:math> of coprime order. Motivated by this result, we study the groups with the property that <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201800403:mana201800403-math-0003" wiley:location="equation/mana201800403-math-0003.png"> <mat:mrow> <mat:msup> <mat:mrow> <mat:mo stretchy="false">(</mat:mo> <mat:mi>x</mat:mi> <mat:mi>y</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> <mat:mi>G</mat:mi> </mat:msup> <mat:mo>=</mat:mo> <mat:msup> <mat:mi>x</mat:mi> <mat:mi>G</mat:mi> </mat:msup> <mat:msup> <mat:mi>y</mat:mi> <mat:mi>G</mat:mi> </mat:msup> </mat:mrow> </mat:math> and those with the property that <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201800403:mana201800403-math-0004" wiley:location="equation/mana201800403-math-0004.png"> <mat:mrow> <mat:mi>χ</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>x</mat:mi> <mat:mi>y</mat:mi> <mat:mo stretchy="false">)</mat:mo> <mat:mo>=</mat:mo> <mat:mi>χ</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>x</mat:mi> <mat:mo stretchy="false">)</mat:mo> <mat:mi>χ</mat:mi> <mat:mo stretchy="false">(</mat:mo> <mat:mi>y</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> </mat:math> for every <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201800403:mana201800403-math-0005" wiley:location="equation/mana201800403-math-0005.png"> <mat:mrow> <mat:mi>χ</mat:mi> <mat:mo>∈</mat:mo> <mat:mo form="prefix">Irr</mat:mo> <mat:mo stretchy="false">(</mat:mo> <mat:mi>G</mat:mi> <mat:mo stretchy="false">)</mat:mo> </mat:mrow> </mat:math> and every nontrivial <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201800403:mana201800403-math-0006" wiley:location="equation/mana201800403-math-0006.png"> <mat:mrow> <mat:mi>x</mat:mi> <mat:mo> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract Type =“Main”XML：Lang =“en”> <标题类型=“main”>抽象</ title> > 最近，Baumslag和Wiegold证明了一个有限的群体 g </ i> 如果才能才能才是 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584X：MANA201800403：MANA201800403-MANA21800403-MATH-0001”WIley：Location =“等式/ MANA201800403-MATH-0001.PNG”> <MAT：MROW> <MAT：MI> O </ MAT：MI> <mat：mo stractry =“false”>（</ mat：mo> <mat：mi> x </ mat：mi> <MAT：MI> Y </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> <mat：mo> = </ mat：mo> <MAT：MI> O </ MAT：MI> <mat：mo stractry =“false”>（</ mat：mo> <mat：mi> x </ mat：mi> <mat：mo stractry =“假”>）</ mat：mo> <MAT：MI> O </ MAT：MI> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MI> Y </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> </ mat：math> 对于每一个 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584X：MANA201800403：MANA201800403-MANA201800403-MATH-0002”WIley：Location =“等式/ MANA201800403-MATH-0002.PNG”> <MAT：MROW> <mat：mi> x </ mat：mi> <mat：mo>，</ mat：mo> <MAT：MI> Y </ MAT：MI> <mat：mo>∈</ mat：mo> <MAT：MI> G </ MAT：MI> </ mat：MROW> </ mat：math> 协调秩序。通过这种结果，我们研究了与属性的群体 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584x：MANA201800403：MANA201800403-MANA201800403-MATH-0003”WIley：位置=“等式/ MANA201800403-MATH-0003.PNG”> <MAT：MROW> <mat：msup> <MAT：MROW> <mat：mo stractry =“false”>（</ mat：mo> <mat：mi> x </ mat：mi> <MAT：MI> Y </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> <MAT：MI> G </ MAT：MI> </ mat：msup> <mat：mo> = </ mat：mo> <mat：msup> <mat：mi> x </ mat：mi> <MAT：MI> G </ MAT：MI> </ mat：msup> <mat：msup> <MAT：MI> Y </ MAT：MI> <MAT：MI> G </ MAT：MI> </ mat：msup> </ mat：MROW> </ mat：math> 和那些有属性的人 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584x：MANA201800403：MANA201800403-MANA201800403-MATH-0004”WIley：位置=“等式/ MANA201800403-MATH-0004.PNG”> <MAT：MROW> <mat：mi>χ</ mat：mi> <mat：mo stractry =“false”>（</ mat：mo> <mat：mi> x </ mat：mi> <MAT：MI> Y </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> <mat：mo> = </ mat：mo> <mat：mi>χ</ mat：mi> <mat：mo stractry =“false”>（</ mat：mo> <mat：mi> x </ mat：mi> <mat：mo stractry =“假”>）</ mat：mo> <mat：mi>χ</ mat：mi> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MI> Y </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> </ mat：math> 对于每一个 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584X：MANA201800403：Mana201800403-Math-0005”Wiley：Location =“等式/ Mana201800403-Math-0005.png”> <MAT：MROW> <mat：mi>χ</ mat：mi> <mat：mo>∈</ mat：mo> <mat：mo form =“prefix”> irr </ mat：mo> <mat：mo stractry =“false”>（</ mat：mo> <MAT：MI> G </ MAT：MI> <mat：mo stractry =“假”>）</ mat：mo> </ mat：MROW> </ mat：math> 而且每一个都是非凡的 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584X：MANA201800403：MANA201800403-MANA21800403-MATH-0006”WIley：Location =“等式/ MANA201800403-MATH-0006.PNG”> <MAT：MROW> <mat：mi> x </ mat：mi> <mat：mo> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" >著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-24615/'>《Mathematische Nachrichten》</a> <b style="margin: 0 2px;">|</b><span>2019年第6期</span><b style="margin: 0 2px;">|</b><span>共6页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Guralnick Robert M.&option=202" target="_blank" rel="nofollow">Guralnick Robert M.;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Moretó Alexander&option=202" target="_blank" rel="nofollow">Moretó Alexander;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Departament of MathematicsUniversity of Southern CaliforniaLos Angeles CA 90089‐2532 USA;</p> <p>Departament de MatemàtiquesUniversitat de València46100 Burjassot València Spain;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li > <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span><a href="https://www.zhangqiaokeyan.com/clc/156.html" title="数学">数学;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=conjugacy class&option=203" rel="nofollow">conjugacy class;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=irreducible character&option=203" rel="nofollow">irreducible character;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=order of elements&option=203" rel="nofollow">order of elements;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=product of elements&option=203" rel="nofollow">product of elements;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=solvable groups&option=203" rel="nofollow">solvable groups;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=20C15–20D15&option=203" rel="nofollow">20C15–20D15;</a> </p> <div class="translation"> 机译：共轭类;不可减少的特征;元素的顺序;元素的产品;可解决的团体;20C15-20D15; </div> </li> </ul> </div> </div> <div class="literature cardcommon" id="literaturereference" style="display:none"> <div class="similarity "> <h3 class="all_title" id="enpatent111">引文网络</h3> <div class="referencetab clearfix"> <ul id="referencedaohang"> <li dataid="referenceul">参考文献</li> <li dataid="citationul">引证文献</li> <li dataid="commonreferenceul">共引文献</li> <li dataid="commoncitationul">同被引文献</li> <li dataid="tworeferenceul">二级参考文献</li> <li dataid="twocitationul">二级引证文献</li> </ul> </div> <div class="reference_details" id="referenceList"> <ul id="referenceul"></ul> <ul id="citationul"></ul> <ul id="commonreferenceul"></ul> <ul id="commoncitationul"></ul> <ul id="tworeferenceul"></ul> <ul id="twocitationul"></ul> </div> </div> </div> <div class="literature cardcommon"> <div class="similarity "> <h3 class="all_title" id="enpatent66">相似文献</h3> <div class="similaritytab clearfix"> <ul> <li class="active" >外文文献</li> <li >中文文献</li> <li >专利</li> </ul> </div> <div class="similarity_details"> <ul > <li> <div> <b>1. </b><a class="enjiyixqcontent" href="/journal-foreign-detail/0704023595726.html">Conjugacy classes, characters and products of elements</a> <b>[J]</b> . <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Guralnick Robert M.&option=202" target="_blank" rel="nofollow" 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href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=陈招斌&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,陈招斌</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=刘盛华&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor">,刘盛华</a> <span> <a href="/conference-cn-22341/" target="_blank" rel="nofollow" class="tuijian_authcolor"> . 第十四次全国电化学会议 </a> <span> <span> . 2007</span> </span> </div> </li> <li> <div> <b>7. </b><a class="enjiyixqcontent" href="/academic-degree-domestic_mphd_thesis/020313735089.html">辽东湾沉积物元素地球化学特征和百年来人类活动的元素记录研究</a> <b>[A] </b> <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=张现荣&option=202" target="_blank" rel="nofollow" class="tuijian_auth tuijian_authcolor"> . 张现荣</a> <span> . 2012</span> </span> </div> </li> </ul> <ul style="display: none;"> <li> <div> <b>1. </b><a class="enjiyixqcontent" href="/patent-detail/06120110931177.html">稀土类元素与铁的分离方法及含稀土类元素熔渣</a> 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