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<html xmlns=<p> question which lies between the Zassenhaus Conjecture and the Prime Graph Question is the Spectrum Problem. It asks, if the orders of elements in <span class="SimpleMath">G</span> and <span class="SimpleMath">mathrmV(ZG)</span> coincide. In general, by a result of J. A. Cohn and D. Livingstone <a href="chapBib.html#biBCohnLivingstone">[CL65]</a>, Corollary 4.1, and a result of M. Hertweck <a href="chapBib.html#biBHertweckSolvable">[Her08a]</a>, the following is known about the possible orders of torsion units in
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<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 expositionjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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 > java.lang.StringIndexOutOfBoundsException: Range [285, 284) out of bounds for length 411
<p>A conjecture ofH..Zassenhausasserted everynormalized of <span java.lang.StringIndexOutOfBoundsException: Range [199, 93) out of bounds for length 753
<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 0,,- 1 ,]<java.lang.StringIndexOutOfBoundsException: Range [104, 103) out of bounds for length 109
<p>A question which lies between the Zassenhaus Conjecture and the Prime Graph Question is the Spectrum Problem. It asks, if the orders of elements in <span class="SimpleMath">G</span> and <span class="SimpleMath">mathrmV(ZG)</span> coincide. In general, by a result of J. A. Cohn and D. Livingstone <a href="chapBib.html#biBCohnLivingstone">[CL65]</a>, Corollary 4.1, and a result of M. Hertweck <a href="chapBib.html#biBHertweckSolvable">[Her08a]</a>, the following is known about the possible orders of torsion units in integral group rings:</p>
<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 if any unit of finite order in <span class="SimpleMath">mathrmV(ZG)</span> is conjugate in the rational group algebra <span class="
<h4p> is clear the Prime Question or 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 <
<p>For a finite group <span class="SimpleMath">G</span> and an element <span class="SimpleMath">x ∈
<p class="pcenter"> \varepsilon_x \colon \mathbb{Z}G \to \mathbb{Z}, \ \ \sum\limits_{g \in G} z_g g \mapsto \sum\limits_{g \in xjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Letclass="u<> atorsionjava.lang.StringIndexOutOfBoundsException: Range [62, 61) out of bounds for length 251
<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">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>From version 4 onwards this package
<><em(<a =".tmlbiBHertweckSolvable><a> 2;<href="htmlbiBHertweckEdinb">[Her08b]</a>, Lemma 2.2; <a href="chapBib.html#biBMargolisHertweck">[Mar17]</a>) Let <span class="SimpleMath">u ∈ mathrmV(ZG)</span> be of finite order and <span class="SimpleMath">ε_g(u) ≠ 0</java.lang.StringIndexOutOfBoundsException: Index 1337 out of bounds for length 1336
<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<>em>Theorem:/em Foraspan =SimpleMath"> ∈ mathrmV(G<span, a group element <span class="Math">s</span>, a prime <span class="Math">p</span> and a natural number <span class="Math">j</span> we have</p>
<p>For more details on when the variable <code class="keyw">HeLP_sol</code> is modified or resetjava.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 conjugacy class of an element <span class="Math">x</span> in <span class="Math">G</span>. Let <span class="Math">u</span> be a torsion unit in <span class="Math">\mathrm{V}(\mathbb{Z}G)</span> of order <spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>for all ordinary characters <span class="Math">\chi</span> and all <span class="Math">l</span>. This formula was given by I.S. Luthar and I.B.S. Passi <a href="chapBib.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="Math">p > 0</span> with Brauer character <span class="Math">\varphi</span>, if <span class="Math">p</span> is coprime to <span class="Math">k</span> <a href="chapBib.html#biBHertweckBrauer">[Her07]</a>, § 4. In that case one has to ignore the <span class="Math">p</span>-singular conjugacy classes (i.e. the classes of elements with an order divisible by <span class="Math">p</span>) and the above formula becomes</p>
<java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 267
p is negativejava.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 85
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>for all Brauer characters <span class="Math">\varphi</span> and all <span class="Math">l</span>.</p>
<>These equations allow to build a system of integral inequalities for the partial augmentations of <span class="Math">u</span>. Solving these inequalities is exactly what the HeLP
<p>For <span class="Math">p</span>-solvable groups representations over fields of characteristic <span class="Math">p</span> can not give any new java.lang.StringIndexOutOfBoundsException: Index 157 out of bounds for length 0
<p>We also included a result motivated by a theorem R. Wagner proved 1995 in his Diplomarbeit <java.lang.StringIndexOutOfBoundsException: Range [0, 96) out of bounds for length 0
<p><em>Theorem:</em> For a torsion unit <span class="java.lang.StringIndexOutOfBoundsException: Index 58 out of bounds for length 0
<p class="pcenter"> \sum\limits_{x^{p^j} \sim s} \varepsilon_x(u) \equiv \varepsilon_sjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>Combining the Theorem with the HeLP-method may only givelispanclass=""> \ A java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 373
<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 java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 5
<p>For the Zassenhaus Conjecture only the following reduction is available:</p>
<p><em>Theorem:</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>It is also known to go over to other types of direct products under certain conditions <a href="chapBib.java.lang.StringIndexOutOfBoundsException: Index 111 out of bounds for length 0
<ul>
<li><p>Nilpotent groups <a href="chapBib.html#biBWeiss91">[Wei91]</a>,</p>
</lijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<li>>Cyclic-By-Abelian groups <a href="chapBib.html#biBCyclicByAbelian">[CMdR13]</a> and some other special cyclic-by-nilpotent groups <a href="chapBib.html#biBCaicedoDelRio20">[CdR20]</a>,</p>
</li>
<li><p>Groupsli><>Solvable groups < href=chapBib.html#biBKimmiPQ">[Kim06]</a>,</p>
</>
<li><p>Frobenius groups whose <li><p>All but two of the sporadic simple and their automorphismgroups<a href="chapBib.biBCaMaBrauerTree>[CM21</>, the exceptions being the Monster and the O'Nan group; for an overview of early HeLP-results see <a href="chapBib.html#biBKonovalovKimmiStAndrews">[KK15]</a>,</p>
</li>
<li><p>Groups <span class="Math">X \rtimes A</span>, where <span class="Math">X</span> and <
</li>
<li><p>All groups of order up to 143 <a href="chapBib.html#biBBaHeKoMaSi">[BHK
</li>
<li><pjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li><p>For special linear groups <span class="Math">SL(2,p)</span> and <span class="Math">SL(2,p^2)</span> for <span class="Math">p</span> a prime <a href="chapBibjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
</ul>
<p>The only known counterexamples to the conjecture are exhibited in <a href="chapBib.html#java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [31, 2) out of bounds for length 144
<p><em>Theorem:</em> Assume the Prime Graph Question holds for all java.lang.StringIndexOutOfBoundsException: Range [0, 73) 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 automorphism group and the whole automorphism java.lang.StringIndexOutOfBoundsException: Index 158 out of bounds for length 0
<ul>
<li><p>Solvable groups <a href="chapBib.html#biBKimmiPQ">[Kim06]</a>,</p>
</li>
<li><p>All but two of the sporadic simple groups and their automorphism groups <a href="chapBib.html#biBCaMaBrauerTree">[CM21]</a>, the exceptions being the Monster and the O'Nan group; for an overview of early HeLP-results see <a href="chapBib.html#biBKonovalovKimmiStAndrews">[KK15]</a>,</p>
</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 class="Math">p</span> denotes a prime, <a href="chapBib.html#biBHertweckBrauer">[Her07]</a>, <a href="chapBib.html#biBBaMa4prI">[BM17a]</a>.</p>
</li>
<li><p>Groups whose socle is isomorphic to an alternating group, <a href="chapBib.html#biBSalimA7A8">[Sal11]</a> <a href="chapBib.html#biBSalimA9A10">[Sal13]</a><a href="chapBib.html#biBBaCa">[BC17]</a><a href="chapBib.html#biBBaMaAn">[BM19a]</a>,</p>
</li>
<li><p>Almost simple groups whose order is divisible by at most three different primes <a href="chapBib.html#biBKonovalovKimmiStAndrews">[KK15]</a> and <a href="chapBib.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.html#biBBaMa4prI">[BM17a]</a><a href="chapBib.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.html#biBCaMaBrauerTree">[CM21]</a></p>
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