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<div class="ChapSects"><a href="chap5_mj.html#X84AF2F1D7D4E7284">5 <span
< classContSect< class"< class"">nbsp;/>5.1 span ="">he Conjecture and questions<>
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</<><emTheorem<em exponents <span="">(\{V}\{Z}G\<span> <span="">\G)/> coincide.Moreover <span="">(\<span, in">(mathrmV(mathbb}G)\) has the same order some element in< class=""\(.)/pan<>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap5_mj.html#X85810FF37EB3F4B4">5.5 <span class
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<h3>5 <span class="Heading"> ="">\xin<s> let class"">\xG\<span theclass span="\(\\G)/pan>.The partialaugmentation respect to< class=""\(\/>orrather class of< =""> )/span sending an element tothesumofthecoefficients elements java.lang.StringIndexOutOfBoundsException: Range [642, 587) out of bounds for length 658
<p><p>et class>u)<span a element < class"\\{}\{}))/span>.By results of G Higman . andM.Hertweckthefollowingis partial augmentations ofof >\u\<span<p
<h4>5.1 <span java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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 of units of augmentation one, aka. normalized units. An element of the unit group of <span class="SimpleMath">\(\mathbb{Z}G\)</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">\(\mathbb{Z}G\)<
<p>Negative solutions to the conjecture were finally found in <a href="chapBib_mj.html#biBEiMa18">[EM18]</a>.</p>
<p><p>The last also the of variablecode class=keyw<> In class">HeLP_solk]<> thepossiblepartial foran element of order \k\<span <span"impleMath>\u^d\)for< class=SimpleMath"\d)/panjava.lang.StringIndexOutOfBoundsException: Range [465, 346) out of bounds for length 682
<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">\(\mathrm{V}(\mathbb{Z}G)\)</span> coincide. In general, by a result of J. A. Cohn and D. Livingstone <a href="chapBib_mj.html#biBCohnLivingstone">[CL65]</a>, Corollary 4.1, and a result of M. Hertweck <a href="chapBib_mj.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="SimpleMath">\(\mathrm{V}(\mathbb{Z}G)\)</span> and <span class="SimpleMath">\(G\)</span> coincide. Moreover, if <span class="SimpleMath">\(G\)</span> is solvable, any torsion java.lang.StringIndexOutOfBoundsException: Range [0, 230) out of bounds for length 0
<p>Finally, a question raised by W. Kimmerle in <a href="chapBib_mj.html#biBAri">[Kim07]</a> asks if any unit of finite order in <span class="SimpleMath">\(\mathrm{V}(\mathbb{Z}G)\)</span> is conjugate in the rational group algebra <span class="SimpleMath">\(\mathbb{Q}H\)</span> to a trivial unit, where <span class="SimpleMath">\(H\)</span> is a finite group containing <span class="SimpleMath">\(G\)</span>. We call this the Kimmerle Problem. This question did not receive much attention while the Zassenhaus Conjecture was still open. It can be shown however that the methods used in <a href
<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">\(x \in G\)</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 <spanclass="SimpleMath">\(x\)</span> is the map <span class="SimpleMath">\(\varepsilon_x \)</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>
<p>Let <span class="SimpleMath">\(u\)</span> be a torsion element in <span class="SimpleMath">\(\mathrm{V}(\mathbb{Z}G)\)</span>. By results of G. Higman, S.D. Berman and M. Hertweck the following is known for the partial augmentations of <span class="SimpleMath">\(u\)</span>:</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) = java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<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 2.5:</p>
<p><em
<> last explains of variable class=keyw<code. In< class"eyw>[k possible augmentationsfor element of order>\()/> dividing classSimpleMathk)/> (except for ="SimpleMath>(dk)/> stored sorted ascendingwrt orderoftheelement spanclass=">\u^d)/>. For , <span="SimpleMath">\(k =1\<span an of ="keyw">[2]/> ight of formp
<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 < class>(x^G\<span class an < classSimpleMath\(\)/="">(\</>. < classSimpleMath(\<span be a torsion in classSimpleMath\\{V(mathbbG)\)</span>of order span=""\k\<spanspan">(\)/pan ordinary representationof< class=SimpleMath>\\)/> overafield contained < class"SimpleMath>\\{})/> character<panSimpleMath(\\<spanspanSimpleMath(())/pan matrix order diagonalizablespan=SimpleMath\\{C})<span Let class">\\\)/> beaprimitive < class=SimpleMath>\k)/>- root unity then themultiplicity < class=SimpleMath\mu_l(u\)\) of of spanclass=SimpleMath>\Du\ computedviaFourier inversion
</li>
<li>p> groups order isdivisible at two different primes< href="chapBib_mj.html#biBJuriaansMilies">[JPM00]</a,</pjava.lang.StringIndexOutOfBoundsException: Index 141 out of bounds for length 141
</li>
<li><p>Groups <span class="SimpleMath">\(X \rtimes A\)</span>, java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
</li>
<li><p>The non-abelian simple groups <span class="
<
<li<>For linear <span="SimpleMath"\(SL(2,p\</> and< class"SimpleMath">\(SL(,p2))<span <span class="SimpleMath">\p\</span> a primeahref"chapBib_mj.html#">[dRS19<a>./p>
</li>
</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>
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