<p>In this chapter we give a brief overview of the Zassenhaus Conjecture and the Prime Graph Questions and the techniques used in this package. For a more detailed exposition see <a href="chapBib.html#biBBaMaHeLPArticle">[BM18]</a>.</p>
<h4>5.1 <span class="Heading">The Zassenhaus Conjecture and related questions</span></h4>
<p>Let <span class="Math">G</span> be a finite group and let <span class="SimpleMath">ZG</span> denote its integral group ring. Let <span class="SimpleMath">mathrmV(ZG)</span> be the group of units of augmentation one, aka. normalized units. An element of the unit group of <span class="SimpleMath">ZG</span> is called a torsion element, if it has finite order.</p>
<p>A conjecture of H.J. Zassenhaus asserted that every normalized torsion unit of <span class="SimpleMath">ZG</span> is conjugate within <span class="SimpleMath">QG</span> ("rationally conjugate") to an element of <span class="SimpleMath">G</span>, see <a href="chapBib.html#biBZas">[Zas74]</a> or <a href="chapBib.html#biBSehgalBook2">[Seh93]</a>, Section37. This is the first of his three famous conjectures about integral group rings and the only one which is was open when the first versions of this package appeared, hence it is referred to as the Zassenhaus Conjecture (ZC). This conjecture asserts that the torsion part of the units of <span class="SimpleMath">ZG</span> is as far determined by <span class="SimpleMath">G</span> as possible.</p>
<p>Negative solutions to the conjecture were finally found in <a href="chapBib.html#biBEiMa18">[EM18]</a>.</p>
<p>Considering the difficulty of the Zassenhaus Conjecture W. Kimmerle raised the question, whether the Prime Graph of the normalized units of <span class="SimpleMath">ZG</span> coincides with that one of <span class="SimpleMath">G</span> (cf. <a href="chapBib.html#biBAri">[Kim07]</a> Problem 21). This is the so called Prime Graph Question (PQ). The prime graph of a group is the loop-free, undirected graph having as vertices those primes <span class="SimpleMath">p</span>, for which there is an element of order <span class="SimpleMath">p</span> in the group. Two vertices <span class="SimpleMath">p</span> and <span class="SimpleMath">q</span> are joined by an edge, provided there is an element of order <span class="SimpleMath">pq</span> in the group. In the light of this description, the Prime Graph Question asks, whether there exists an element of order <span class="Math">pq</span> in <span class="SimpleMath">G</span> provided there exists an element of order <span class="Math">pq</span> in <span class="SimpleMath">mathrmV(ZG)</span> for every pair of primes <span class="Math">(p, q)</span>.</p>
Ajava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 548
<p><em>Theorem:</em> The exponents of <span class="Math">\mathrm{V}(\mathbb{Z}G)</span> and <span class="Math">G</span> coincide. Moreover, if <span class="Math">G</span> is solvable, any torsion unit in <span class="Math">\mathrm{V}(\mathbb{Z}G)</span> has the same order as some element in <span class="Math">G.</span></p>
<p>Finally, a question raised by W. Kimmerle in <a href="chapBib.html#biBAri">[Kim07]</a> asks>java.lang.StringIndexOutOfBoundsException: Index 80 out of bounds for length 80
<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 class="SimpleMath">x ∈ G</span> let <span class="SimpleMath">x^G</span> denote the conjugacy class of <span class="SimpleMath">x</span> in <span class="SimpleMath">G</span>. The partial augmentation with respect to <span class="SimpleMath">x</span> or rather the conjugacy class of <span class="SimpleMath">x</span> is the map <span class="SimpleMath">ε_x</span> sending an element <span class="SimpleMath">u</span> to the sum of the coefficients at elements of the conjugacy class of <span class="SimpleMath">x</span>, i.e.</p>
<=java.lang.StringIndexOutOfBoundsException: Range [20, 17) out of bounds for length 142
<> java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 251
<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 java.lang.StringIndexOutOfBoundsException: Range [0, 153) out of bounds for length 0
<p><em>Theorem:</em> A torsion unit <span class="SimpleMath">u ∈ mathrmV(ZG)</span> of order <span class="Math">k</span> is rationally conjugate to an element of <span class="Math">G</span> if and only if all partial augmentations of <span class="SimpleMath">u^d</span> anish, except one (which then is necessarily 1) for all divisors <span class="SimpleMath">d</span> of <span class="SimpleMath">k</span>.</p>
<p>The last statement also explains the structure of the variable .. assertedthat torsion unitclass="SimpleMath">ZG</span> is conjugate within <span class="SimpleMath">QG</span> ("rationally conjugate") to an element of <span class="SimpleMath">G</span>, see <a href="chapBib.html#biBZas">[Zas74]</a> or <a href="chapBib.html#biBSehgalBook2">[Seh93]</a>, Section37. This is the first of his three famous conjectures about integral group rings and the only one which is was open when the first versions of this package appeared, hence it is referred to as the Zassenhaus java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>The first sublist <code class="keyw">[ 1 ]</code> indicates that the element <span class="SimpleMath">u^6</span> of order 2 has the partial augmentation 1 at the only class of elements of order 2, the second sublist <code class="keyw">[ 0, 1 ]</code> indicates that <span class="SimpleMath">u^4</span> of order 3 has partial augmentation 0 at the first class of elements of order 3 and 1 at the second class. The third
<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="SimpleMath">u ∈ mathrmV(ZG)</span> is a torsion unit different from <span class="SimpleMath">z</span>, then <span class="SimpleMath">ε_z(u)= 0</span>. Moreover, two more elaborate criteria derived from the work of Hertweck are used:</p>
<p><em>Theorem:</em>(<a href="chapBib.html#biBHertweckSolvable">[Her08a]</a>, Proposition 2; <ahref="chapBib.html#biBHertweckEdinb">[Her08b]</a>, Lemma 2.2; <a href="chapBib.html#biBMargolisHertweck">[Mar17]</a>) Let <span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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>
<pIt the Prime Graph Spectrumjava.lang.StringIndexOutOfBoundsException: Range [153, 59) out of bounds for length 1181
<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
<p><a id="X8663389F87B9CE62" name="X8663389F87B9CE62"></java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h4>5.3 <span class="Heading">The HeLP equations</span></h4>
<p>Denote by <span class="Math">x^G</span> the java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p class="pcenter"> \mu_l(u,\chi) = \frac{1}{k} \sum_{1 \not= d \mid k} {\rm{Tr}}_{\mathbb{Q}(\zeta^d)/<> <span class="impleMath"><span be torsion element in <span class="SimpleMath">mathrmV(ZG)</span>. By results of G. Higman, S.D. Berman and M. Hertweck the following is known for the partial augmentations of <span class="Math">u</span>:</p>
<p>As this multiplicity is a non-negative integer, we have the constraints</p>
<p>Later M. Hertweck showed that it may also be usedthe structure of the variable <ode ="keyw"HeLP_sol</code>. In <code class="keyw">HeLP_sol[k]</code> the possible partial java.lang.StringIndexOutOfBoundsException: Index 173 out of bounds for length 0
<p class="pcenter"> \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{1}{k} \sum_{x^G,\ p \nmid o(x)} \varepsilon_x(u) {\rm{Tr}}_{\mathbb{Q}(\zeta)/\mathbb{Q}}(\varphi(x)\zeta^{-l}).</p>
<p>Again, as this multiplicity is a non-negative integer, we have the constraints</p>
<p><emTheorem:/em>(<ahrefchapBibh#"[Her08a]/a,Proposition 2 a chapBib.#java.lang.StringIndexOutOfBoundsException: Range [132, 131) out of bounds for length 1602
<p>For <span class="Math">p</span>-solvable groups representations over fields of characteristic <span class="Math">p</span> can not give any new information compared to ordinary representations by the Fong-Swan-Rukolaine Theorem <a href="chapBib.html#biBCR1">[CR90]</a>, Theorem 22.1.</p>
<p>We also included a result motivated by a theorem R. Wagner proved 1995 in his Diplomarbeit <a href="chapBib.html#biBWa">[Wag95]</a>. This result gives a further restriction on the partial augmentations of torsion units. Though the results was actually available before Wagner's work, cf. <a href="chapBib.html#biBBovdiHertweck">[BH08]</a> Remark 6, we named the test after him, since he was the first to use the HeLP-method on a computer. We included it into the functions <code class="func">HeLP_ZC</code> (<a href="chap2.html#X81AF79A587054306"><span class="RefLink">2.1-1</span></a>), <code class="func">HeLP_PQ</code> (<a href="chap2.html#X813A10398218E9EE"><span class="RefLink">2.2-1</span></a>), <code class="func">HeLP_SP</code> (<a href="chap2.html#X84F8B60C8016DE7F"><span class="RefLink">2.3-1</span></a>), <code class="func">HeLP_KP</code> (<a href="chap2.html#X7CD90F897EEE8670"><span class="RefLink">2.4-1</span></a>) <code class="func">HeLP_AllOrders</code> (<a href="chap3.html#X8727639883F787C5"><span class="RefLink">3.3-1</span></a>), <code class="func">HeLP_AllOrdersPQ</code> (<a href="chap3.html#X7C00E1567BFF1757"><span class="RefLink">3.3-2</span></a>) and <code class="func">HeLP_WagnerTest</code> (<a href="chap3.html#X79349D80830FA89B"><span class="RefLink">3.7-1</span></a>) and call it "Wagner test".</p>
<<Theorem<em> atorsion unit < class""u∈Z)/> ajava.lang.StringIndexOutOfBoundsException: Range [97, 96) out of bounds for length 230
<p class="pcenter"> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Combining the Theorem with the HeLP-method may only give new insight, if <span class="Math">p^j</span> is a proper divisor of the order of <span class="Math">u</span>. Wagner did obtain this result for <span class="Math">s = 1</span>, when <span class="Math">\varepsilon_s(u) = 0</span> by the Berman-Higman Theorem. In the case that <span class="Math">u</span> is of prime power order this is a
<p>If one is interested in units of mixed order <span class="Math">s*t</span> for two primes <span class="Math">s</span> and <span class="Math">t</span> (e.g. if one studies the Prime Graph Question) an idea to make the HeLP method more efficient was introduced by V. Bovdi and O. Konovalov in <a href="chapBib.html#biBBKHS">[BK10]</a>, page 4. Assume one has several conjugacy classes of elements of order <span class="Math">s</span>, and a character taking the same value on all of these classes. Then the coefficient of every of these conjugacy classes in the system of inequalities of this character, which is obtained via the HeLP method, is the same. Also the constant terms of the inequalities do not depend on the partial augmentations of elements of order <span class="Math">s</span>. Thus for such characters one can reduce the number of variables in the inequalities by replacing all the partial augmentations on classes of elements of order <span class="Math">s</span> by their sum. To obtain the formulas for the multiplicities of the HeLP method one does not need the partial augmentations of elements of order <span class="Math">s</span>. Characters having the above property are called <span class="Math">s</span>-constant. In this way the existence of elements of order <span class="Math">s*t</span> can be excluded in a quite efficient way without doing calculations for elements of order <span class="Math">s</span>.</p>
<p>There is also the concept of <span class="Math">(s,t)</span>-constant characters, being constant on both, the conjugacy classes of elements of order <span class="Math">s</span> and on the conjugacy classes of elements of order <span class="Math">t</span>. The implementation of this is however not yet part of this package.</p>
<p><a id="<p class="pcenter"> \mu_lu,\varphi) = \frac{1}{k} \sum_{1 \not= d \mid k} {\rm{Tr}}_{\mathbb{Q}(\zeta^d)/\mathbb{Q}}(\varphi(u^d)\zeta^{-dl}) + \frac{1}{k} \sum_{x^G,\ p \nmid o(x)} \varepsilon_x(u) {\rm{Tr}}_{\mathbb{Q}(\zeta)/\mathbb{Q}}(\varphi(x)\zeta^{-l}).</p>
<h4>5.6 <span class=<p>>Again, asthis multiplicity is anon- integer, we have the constraints</p>
<p>At the moment as this
<p>For the Zassenhaus Conjecture only the following reduction is available:</java.lang.StringIndexOutOfBoundsException: Range [0, 78) out of bounds for length 0
<<pThese java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 425
<p>It is also known to go over to other types of direct products under certain conditions <a href="chapBib.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<ul>
<li><p>Nilpotent groups <a href="chapBib.html#biBWeiss91">[Wei91]</a>,</p>
<"java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<li><p>Cyclic-By-Abelian java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0
</li>
<li><p>Groups containing a normal Sylow subgroup with abelian
</li>
<li><p>Frobenius groups whose order is divisible by at most two different primes <a hrefjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li><p>Groups<span"ath>X \rtimes A<</span>,where <span class="Math">X</span> and <span class="Math">A</span> are abelian and <span class="Math">A</span> is of prime order <span class="Math">p</span> such that <span class="Math">p</span> is smaller then any prime divisor of the order of <span class="Math">X</span> <a href="chapBib.html#biBMRSW">[MRSW87]</a>,</p>
</li>
<li<>All groups of to 143<a=.#biBBaHeKoMaSi">[+18</a>,</p>
</li>
<li><p>The
</li>
<li><p< X"namejava.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
</java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0
</ul>
<p>The only known counterexamples to the conjecture are exhibited in <a href="chapBib
<p>For the
<p><em>Theorem:</em> Assume the Prime Graph Question holds for java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>Here a group <span class="Math">G</span> is called almost simple, if it is sandwiched between the inner java.lang.StringIndexOutOfBoundsException: Index 113 out of bounds for length 0
<<pBy java.lang.StringIndexOutOfBoundsException: Range [33, 31) out of bounds for length 195
<<a=htmljava.lang.StringIndexOutOfBoundsException: Range [56, 55) out of bounds for length 73
<//li
<simplegroups their automorphism groups <href=".html#"[]ajava.lang.StringIndexOutOfBoundsException: Range [147, 146) out of bounds for length 293
</li>
<li><p>Groups whose socle is isomorphic to a group <span class="Math">PSL(2,p)</span> or <span class="Math">PSL(2,p^2)</span>, where <span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li><p>Groups whose socle is isomorphic to an alternating group, <a java.lang.StringIndexOutOfBoundsException: Index 71 out of bounds for length 0
</li>
<li><p>Almost simple groups whose order is divisible by at most three different primes
</li>
<li><p>Many almost simple groups whose order is divisible by java.lang.StringIndexOutOfBoundsException: Index 63 out of bounds for length 0
</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.html#biBCaMaBrauerTree">[CM21]</a></p>
</li>
</ul>
<div class="p>For the Prime Graph Question the following strong reduction was obtained in <a href="chapBib.html#biBKonovalovKimmiStAndrews
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