<p id="mathjaxlink" class="pcenter"><a href="chap5.html">[MathJax off]</a></p>
<p><a id="X84AF2F1D7D4E7284" name="X84AF2F1D7D4E7284"></a></p>
<div class="ChapSects"><a href="chap5_mj.html#X84AF2F1D7D4E7284">5 <span class="Heading">Background</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7B02D6AE80303BEB">5.1 <span class="Heading">The Zassenhaus Conjecture and related questions</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7E2BFEC182B09895"><
<span
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X8663389F87B9CE62">5.3 <span class="Heading">The HeLP equations</span<ajava.lang.StringIndexOutOfBoundsException: Index 174 out of bounds for length 174
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X7BA77C9F86ADD546">5.4 <span class="Heading">The Wagner test</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X85810FF37EB3F4B4">5.5 <span class="Heading">s-constant characters</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X79BE759E7F35150E">5.6 <span class="Heading">Known results about the Zassenhaus Conjecture and the Prime Graph Question</span></a>
</span>
</div>
</div>
<h3>5 <span class="Heading">Background</
<p>Inthis we give a brief overview theZassenhausConjecture and PrimeGraph and techniquesused thispackageFor amoredetailed exposition see <a href="chapBib_mj.html#biBBaMaHeLPArticle">[BM18]</a>.</p>
<h4>5.1 <span class="Heading">The Zassenhaus Conjecture and related questions</span></h4>
<p>Let <span class="SimpleMath">\(G\)</span> be a finite group and let <span class="SimpleMath">\(\mathbb{Z}G\)</span> denote its integral group ring. Let <span class="SimpleMath">\(\mathrm{V}(\mathbb{Z}G)\)</span> be the group java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Negative solutions to the conjecture were finally found in <a href="chapBib_mj.html#biBEiMa18">[EM18]</a>.</p>
<p>Considering the difficulty of the Zassenhaus Conjecture W. Kimmerle<div
<p>A question>
p<>Theorem:/> The of classSimpleMath\\athrm}(mathbb))/spanand classSimpleMath\(\</spancoincide Moreover,if class"SimpleMath>\G\)/> is solvable anytorsionunitin \\{}\{Z/panhassameaselement span="SimpleMath>(G\
<p><a id=/spanjava.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
<h4>5.2 <span class="Heading">Partial augmentations and the structure of HeLP sol</span></h4>
<p>For a finite group <span class="SimpleMath">\(G\)</span> and an element <span classSimpleMath( \ G\)</pan <span=SimpleMath(^G)/> denote conjugacy of< class=SimpleMath>(x)/>in classSimpleMath>(\</ with span"SimpleMath>x\SimpleMath>\(x\)</span> is the map <span class="SimpleMath">\(\varepsilon_x\)</>sendingelementspan=SimpleMath>u<span atof the conjugacy class of <span class="SimpleMath">\(x\)</span>, i.e.</p>
<p> <span ="SimpleMath"\(\<span> betorsion in span="SimpleMath>(\mathrmV(mathbbZG\ By results G.Higman,S..Bermanand M knownforthepartial of SimpleMath()/>:/p>
<p><em>Theorem:</em> (<a href="chapBib_mj.html#biBSehgalBook2">[Seh93]</a>, Proposition (1.4); <a href="chapBib_mj.html#biBHertweckBrauer">[Her07]</a>, Theorem 2.3) <span class="SimpleMath">\(\varepsilon_1(u) = 0\)</span> if <span class="SimpleMath">\(u \not= 1\)</span> and <span class="SimpleMath">\(\varepsilon_x(u) = 0\)</span> if the order of <span class="SimpleMath">\(x\)</span> does not divides the order of <span class="SimpleMath">\(u\)</span>.</p>
<p>Partial augmentations are connected to (ZC) and (PQ) via the following result, which is due to Z. Marciniak, J. Ritter, S. Sehgal and A. Weiss <a href="chapBib_mj.html#biBMRSW">[MRSW87]</a>, Theorem <
<p><em>Theorem:</em> A torsion unit <span class="SimpleMathjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< statementalsoexplains structure the <code class=">HeLP_solIn (k)/> and all powers (^d\)>\(k\)</span> (except for <span class="SimpleMath">\(d=k\)</span>) are stored, sorted ascending w.r.t. order of the element <span class="SimpleMath">\(u^d\)</span>. For instance, for <span class="SimpleMath">\(k = 12\)</span> an entry of <code class="keyw">HeLP_sol[12]</code> might be of the following form:</p>
<p>The first sublist <code class="keyw">[ 1 ]</code> indicates that the element <span class="SimpleMath">\(u^6\)<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>From version 4 onwards this package incorporates more theoretical restrictions on partial augmentations. More precisely, it uses more results about vanishing partial augmentations of normalized torsion units. One is the more general form of the Berman-Higman theorem, namely that if <span class="SimpleMath">\(z\)</span> is a central element in <span class="SimpleMath">\(G\)</span> and <span class=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><em>Theorem:</em>(<a href="chapBib_mj.html#biBHertweckSolvable">[Her08a]</a>, Proposition 2; <a href="chapBib_mj.html#biBHertweckEdinb">[Her08b]</a>, Lemma 2.2; <a href="chapBib_mj.html#biBMargolisHertweck">[Mar17]</a>) Let <span class="SimpleMath">\(u \in \mathrm{V}(\mathbb{Z}G)\)</span> be of finite order and <span class="SimpleMath">\(\varepsilon_g(u) \neq 0\)</span> for some <span class="SimpleMath">\(g \in G\)</span>. Suppose that <span class="SimpleMath">\(u\)</span> has smaller order modulo some normal <span class="SimpleMath">\(p\)</span>-subgroup <span class="SimpleMath">\(N\)</span> of <span class="SimpleMath">\(G\)</span>. Then the <span class="SimpleMath">\(p\)</span>-part of <span class="SimpleMath">\(g\)</span> has the same order as the <span class="SimpleMath">\(p\)</span>-part of
<p>To apply this theorem, some knowledge on the normal subgroups of <span class="SimpleMath">\(G\)</span> is necessary. Hence it is only applied in the package when the character table one works with possesses an underlying group.</p>
<p>It is clear the Prime Graph Question or Spectrum Problem can be studied using the HeLP-method (if no possible partial augmentations exist for a given order neither does a unit of that order) and the possibility to do this for the Zassenhaus Conjecture is given via the above theorem of Marciniak-Ritter-Sehgal-Weiss. For the Kimmerle Problem a somehow similar result states that a unit <span class="SimpleMath">\(u \in \mathrm{V}(\mathbb{Z}G)\)</span> of order <span class="SimpleMath">\(k\)</span> is conjugate in <span class="SimpleMath">\(\mathbb{Q}H\)</span>, for <span class="SimpleMath">\(H\)</span> some group containing <span class="SimpleMath">\(G\)</span>, to a trivial unit if and only if thejava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>For more details on when the variable <code class="keyw">HeLP_sol</code> is modified or reset and how to influence this behavior see Section <a href="chap4_mj.html#X84ED1F0D7A47B055"><span class="RefLink">4.2</span></a> and <code class="func">HeLP_ChangeCharKeepSols</code> (<a href="chap3_mj.html#X7BB9009482784E90"><span class="RefLink">3.4-1</span></a>).</p>
<h4>pThe statement also thestructure the <codeclass"">HeLP_sol/>.In codeclass=""HeLP_sol]/>thepartial anof <span=SimpleMath(\<span allpowers span="SimpleMath>ud\/>forSimpleMath\(d\<span <span="">\(\)<spanexcept <spanclass"\d=\>\u^\<span Forinstance forspan classSimpleMath 2)/span> entry <codeclasskeywHeLP_sol1]<codem be thefollowing:</>
<p>Denote by<span="SimpleMath"\()/span> theconjugacyclass of elementspan="SimpleMath">\(<span> in <span classSimpleMath\G)<span Letspan="SimpleMath">\(u)/>torsion unit <span="SimpleMath">\(mathrm}\{Z} oforder < class"impleMath>\(k)/> and < class=SimpleMath"\D\<s>an span""\G)<span fieldinspan=SimpleMath">\(mathbbC\\(chi)/>. Then < class="">\(u\ isamatrix of finite order and thusdiagonalizable over ">(mathbbC\/>. Let \(\<spanthof,then span"">\(\mu_l,chi)</pan span"impleMath>\zeta^)> asaneigenvalueof < "">(())/pan>canbe Fourier inversionandequals/>
ass[\u_lu,chi \rac1{}\sum_ = d mid{rm}}_\{Q}\^d\mathbb(\(u^d)zetad} +\{1}{}\sum_{varepsilon_x(){rm}}{mathbb\)/\{Q}}\(x)\zeta^{-l)\<p
<p>As this
<p class="> more details on when the variable < ="keywHeLP_sol isor nd influence behaviorSection =chap4_mj.#X84ED1F0D7A47B055><span class="java.lang.StringIndexOutOfBoundsException: Range [207, 206) out of bounds for length 364
<p>for all ordinary characters <span class="SimpleMath">\(\chi\)</span> and all <span class="SimpleMath">\(l\)</span>. This formula was given by I.S. Luthar and I.B.S. Passi <a href="chapBib_mj.html#biBLP">[LP89]</a>.</p>
<p>Later M. Hertweck showed that it may also be used for a representation over a field of characteristic <span class="SimpleMath">\(p > 0\)</span> with Brauer character <span class="SimpleMath">\(\varphi\)</span>, if <span class="SimpleMath">\(p\)</span> is coprime to <span class="SimpleMath">\(k\)</span> <a href="chapBib_mj.html#biBHertweckBrauer">[Her07]</a>, § 4. In that case one has to ignore the <span class="SimpleMath">\(p\)</span>-singular conjugacy classes (i.e. the classes of elements with an order divisible by <span class="SimpleMath">\(p\)</span>) and the above formula becomes</p>
<p class="center">\[ \mu_l(u,\varphi) = \frac{1}{k} \sum_{1 \not= d \mid k} {\rm{Tr}}_{\mathbb{Q}(\zeta^d)/\mathbb{Q}}(\varphi(u^d)\zeta^{-dl}) + \frac{<pDenoteby="\xG)< ="SimpleMath">\(x\) in \(G\). Let \u\beatorsionunitin"(mathrmV}({Z})\>(\<spanspan="ordinary < ="impleMath\G)/spanover contained s =""(mathbb)/> =""\chi/>. < =SimpleMath(u\)<pan is of thus< =SimpleMath\(\{C\)</spanLetspan=""(zetaspan < =SimpleMath\(\)/>-throot ,thenmultiplicity =SimpleMath>(\(u,\chi)\)/span < class"">(zeta\)/> as eigenvalue <span class"SimpleMath">\(())/> canbe computed Fourierinversion and </p
<p>Again, as this multiplicity is a non-negative integer
<p>These equations allow to build a system of integral inequalities for the partial augmentations of <span class="SimpleMath">\(u\)</span>. Solving these inequalities is
<p>For <span class="SimpleMath">\(p\)</span>-solvable groups representations over fields of characteristic <span class="SimpleMath">\(p\)</span> can not give any new information compared to ordinary representations by the Fong-Swan-Rukolaine Theorem <a href="chapBib_mj.html#biBCR1">[CR90]</a
<p>We also java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<
<p p><em>Theorem:<em t unit ="SimpleMath"\u \ \{}(mathbb})\/> agroup span="">\s)spanprimeclass"\(\/span a span class""\(\we have
<p>Combining the Theorem with the HeLP-method may only
<p><a id
<h4>>a=X85810FF37EB3F4B4"X85810FF37EB3F4B4pjava.lang.StringIndexOutOfBoundsException: Index 62 out of bounds for length 62
mixed <"impleMath>(st)/> span =""\s)/> < class"impleMath\\<spane.g onePrime) make methodefficientintroducedV. O. ahref.tml#biBBKHS">[BK10] 4.Assume one several of oforder < class="">(\, acharactertaking samevalueonallof classes. the coefficientof classes in thesystem ofinequalities of this character whichis via HeLP method,is thesame Also the constant terms the do not depend the partial augmentationsofelementsoforder< class""\()s\<>by sum.Tothe he multiplicitiesof one not thepartialaugmentationsof order>s)/panCharacterstheproperty <span""\s\<>-. order =SimpleMath\s*\</> excludedquite way < classSimpleMath\s)/><p
<p>There is also the concept of <span class="SimpleMath">\((s,t)\)</span>-constant characters, being constant on both, the conjugacy classes of elements of order <span class="SimpleMath">\(s\)</span> and on the conjugacy classes of elements of order <span class="SimpleMath">\(t\)</span>. The implementation of this is however not yet part of this package.</p>
<h4>5.6 <span class="Heading">Known results about the Zassenhaus Conjecture and the Prime Graph Question</span></h4>
<p>At the moment as this documentation was written, to the best of our knowledge, the following results were available for the Zassenhaus Conjecture and the Prime Graph Question:</p>
<p>For the Zassenhaus Conjecture only the following reduction is available:</p>
<p><em
<p>It is>or Zassenhaus following :<p
<ul also go types products conditions =".html#">[BKS20/>. reductions mindZassenhaus known:/p
<li
</li>
<lijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li
</li><For special groups class>SL))spanspan=SimpleMathSL2^\</>for class>(\)/span < =biBdelRioSerrano19]/>.<pjava.lang.StringIndexOutOfBoundsException: Index 238 out of bounds for length 238
</ul>
<p>The only known counterexamples to the conjecture are exhibited in <a href="chapBib_mj.html#biBEiMa18">[EM18]</a>.</p>
<p>For the Prime Graph Question the following strong reduction was obtained in <a href="chapBib_mj.html#biBKonovalovKimmiStAndrews">[KK15]</a>:</p>
<p><em>Theorem:</em> Assume the Prime Graph Question holds for all almost simple images of a group <span class="SimpleMath">\(G\)</span>. Then (PQ) also holds for <span class="SimpleMath">\(G.\)</span></p>
<p>Here a group <span class="SimpleMath">\(G\)</span> is called almost simple, if it is sandwiched between the inner automorphism group and the whole automorphism group of a non-abelian simple group <span class="SimpleMath">\(S\)</span>. I.e. <span class="SimpleMath">\(Inn(S) \leq G \leq Aut(S).\)</span> Keeping this reduction in mind (PQ) is known for:</p>
<ul>
<li><p>Solvable groups <a href="chapBib_mj.html#biBKimmiPQ">[Kim06]</a>,</p>
</li>
<li><p>All but two of the sporadic simple groups and their automorphism groups <a href="chapBib_mj.html#biBCaMaBrauerTree">[CM21]</a>, the exceptions being the Monster and the O'Nan group; for an overview of early HeLP-results see [KK15],
</li>
<li><p>Groups whose socle is isomorphic to a group <span class="SimpleMath">\(PSL(2,p)\)</span> or <span class="SimpleMath">\(PSL(2,p^2)\)</span>, where <span class="SimpleMath">\(p\)</span> denotes a prime, <a href="chapBib_mj.html#biBHertweckBrauer">[Her07]</a>, <a href="chapBib_mj.html#biBBaMa4prI">[BM17a]</a>.</p>
</li>
<li><p>Groups whose socle is isomorphic to an alternating group, <a href="chapBib_mj.html#biBSalimA7A8">[Sal11]</a> <a href="chapBib_mj.html#biBSalimA9A10">[Sal13]</a><a href="chapBib_mj.html#biBBaCa">[BC17]</a><a href="chapBib_mj.html#biBBaMaAn">[BM19a]</a>,</p>
</li>
<li><p>Almost simple groups whose order is divisible by at most three different primes <a href="chapBib_mj.html#biBKonovalovKimmiStAndrews">[KK15]</a> and <a href="chapBib_mj.html#biBBaMaM10">[BM17b]</a>. (This implies that it holds for all groups with an order divisible by at most three primes, using the reduction result above.)</p>
</li>
<li><p>Many almost simple groups whose order is divisible by four different primes <a href="chapBib_mj.html#biBBaMa4prI">[BM17a]</a><a href="chapBib_mj.html#biBBaMa4prII">[BM19b]</a>,</p>
</li>
<li><p>Certain infinite series of simple groups of Lie type of small rank and other groups from the character table library <a href="chapBib_mj.html#biBCaMaBrauerTree">[CM21]</a></p>
¤ Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.0.22Bemerkung:
¤
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
