Quelle purpose.autodoc Sprache: unbekannt

@Chapter Background

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 <Cite Key="BaMaHeLPArticle"/>.

@Section The Zassenhaus Conjecture and related questions

Let $G$ be a finite group and let <M>\mathbb{Z}G</M> denote its
integral group ring. Let <M>\mathrm{V}(\mathbb{Z}G)</M> be the
group of units of augmentation one, aka. normalized units. An
element of the unit group of <M>\mathbb{Z}G</M> is called a torsion
element, if it has finite order. 

A conjecture of H.J. Zassenhaus asserted that every
normalized torsion unit of <M>\mathbb{Z}G</M> is conjugate within
<M>\mathbb{Q}G</M> ("rationally conjugate") to an element of <M>G</M>, see
<Cite Key="Zas"/> or <Cite Key="SehgalBook2"/>, Section 37. 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 <M>\mathbb{Z}G</M> is as far
determined by <M>G</M> as possible. 

Negative solutions to the conjecture were finally found in <Cite Key="EiMa18"/>. 

Considering the difficulty of the Zassenhaus Conjecture W. Kimmerle raised the
question, whether the Prime Graph of the normalized units of
<M>\mathbb{Z}G</M> coincides with that one of <M>G</M>
(cf. <Cite Key="Ari"/> 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 <M>p</M>, for which there
is an element of order <M>p</M> in the group. Two vertices <M>p</M> and
<M>q</M> are joined by an edge, provided there is an element of order <M>pq</M>
in the group. In the light of this description, the Prime Graph Question asks,
whether there exists an element of order $pq$ in <M>G</M> provided there exists an element of order
$pq$ in <M>\mathrm{V}(\mathbb{Z}G)</M> for every pair of primes $(p, q)$. 

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 <M>G</M> and <M>\mathrm{V}(\mathbb{Z}G)</M>
coincide.
In general, by a result of J. A. Cohn and D. Livingstone <Cite Key="CohnLivingstone"/>,
Corollary 4.1, and a result of M. Hertweck <Cite Key="HertweckSolvable"/>, the
following is known about the possible orders of torsion units in integral group rings: 
**Theorem:** The exponents of $\mathrm{V}(\mathbb{Z}G)$ and $G$ coincide. Moreover, if
$G$ is solvable, any torsion unit in $\mathrm{V}(\mathbb{Z}G)$ has the same order as
some element in $G.$ 

Finally, a question raised by W. Kimmerle in <Cite Key="Ari"/> asks if any unit of finite order
in <M>\mathrm{V}(\mathbb{Z}G)</M> is conjugate in the rational group algebra <M>\mathbb{Q}H</M>
to a trivial unit, where <M>H</M> is a finite group containing <M>G</M>. 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 <Cite Key="EiMa18"/> to construct
counterexamples to the Zassenhaus Conjecture can not yield negative solutions to the Kimmerle Problem.
In this sense it remains the strongest statement about torsion units in integral group rings of finite
group which could still be true.

@Section Partial augmentations and the structure of HeLP_sol

For a finite group <M>G</M> and an element <M>x \in G</M> let <M>x^G</M> denote
the conjugacy class of <M>x</M> in <M>G</M>. The partial augmentation with
respect to <M>x</M> or rather the conjugacy class of <M>x</M> is the map
<M>\varepsilon_x </M> sending an element
<M>u</M> to the sum of the coefficients at elements of
the conjugacy class of <M>x</M>, i.e.
$$ \varepsilon_x \colon \mathbb{Z}G \to \mathbb{Z}, \ \ \sum\limits_{g \in G} z_g g \mapsto \sum\limits_{g \in x^G} z_g. $$

Let <M>u</M> be a torsion element in <M>\mathrm{V}(\mathbb{Z}G)</M>. By results of G. Higman, S.D. Berman and M. Hertweck the following is known for the partial augmentations of $u$: 
**Theorem:** (<Cite Key="SehgalBook2"/>, Proposition (1.4); <Cite Key="HertweckBrauer"/>, Theorem 2.3)
$\varepsilon_1(u) = 0$ if $u \not= 1$ and $\varepsilon_x(u) = 0$ if the order of $x$ does not divides the order of $u$.

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
<Cite Key="MRSW"/>, Theorem 2.5: 
**Theorem:** A torsion unit <M>u \in \mathrm{V}(\mathbb{Z}G)</M> of order $k$
is rationally conjugate to an element of $G$ if and only if all partial augmentations
of <M>u^d</M> vanish, except one (which then is necessarily 1) for all divisors
<M>d</M> of <M>k</M>.

The last statement also explains the structure of the variable <K>HeLP_sol</K>. In <K>HeLP_sol[k]</K>
the possible partial augmentations for an element of order $k$ and all powers <M>u^d</M>
for <M>d</M> dividing <M>k</M> (except for <M>d=k</M>) are stored, sorted ascending w.r.t. order of the
element <M>u^d</M>. For instance, for <M>k = 12</M> an entry of <K>HeLP_sol[12]</K> might be
of the following form:
<K>[ [ 1 ],[ 0, 1 ],[ -2, 2, 1 ],[ 1, -1, 1 ],[ 0, 0, 0, 1, -1, 0, 1, 0, 0 ] ]</K>.
The first sublist <K>[ 1 ]</K> indicates that the element <M>u^6</M> of order 2 has the
partial augmentation 1 at the only class of elements of order 2, the second sublist
<K>[ 0, 1 ]</K> indicates that <M>u^4</M> of order 3 has partial augmentation 0 at the first
class of elements of order 3 and 1 at the second class. The third sublist <K>[ -2, 2, 1 ]</K>
states that the element <M>u^3</M> of order 4 has partial augmentation -2 at the class
of elements of order 2 while 2 and 1 are the partial augmentations at the two classes of elements of order 4 respectively,
and so on. Note that this format provides all necessary information on the partial augmentations
of <M>u</M> and its powers by the above restrictions on the partial augmentations.

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 <M>z</M> is a central element in <M>G</M> and
<M>u \in \mathrm{V}(\mathbb{Z}G)</M> is a torsion unit different from <M>z</M>, then <M>\varepsilon_z(u)= 0</M>.
Moreover, two more elaborate criteria derived from the work of Hertweck are used:
**Theorem:**(<Cite Key="HertweckSolvable"/>, Proposition 2; <Cite Key="HertweckEdinb"/>, Lemma 2.2;
<Cite Key="MargolisHertweck"/>) Let <M>u \in \mathrm{V}(\mathbb{Z}G)</M> be of finite order and <M>\varepsilon_g(u) \neq 0</M>
for some <M>g \in G</M>.
Suppose that <M>u</M> has smaller order modulo some normal <M>p</M>-subgroup <M>N</M> of <M>G</M>.
Then the <M>p</M>-part of <M>g</M> has the same order as the <M>p</M>-part of <M>u</M>. Furthermore, if the <M>p</M>-part
of <M>u</M> is <M>p</M>-adically conjugate to an element in <M>G</M>, then the <M>p</M>-part of <M>g</M> is even
conjugate in <M>G</M> to the <M>p</M>-part of <M>u</M>. Such a <M>p</M>-adic conjugation holds, if the order of <M>u</M>
modulo a normal <M>p</M>-subgroup of <M>G</M> is not divisible by <M>p</M>, i.e. the $p$-part of $u$ is trivial modulo a normal $p$-subgroup. 
To apply this theorem, some knowledge on the normal subgroups of <M>G</M> is necessary. Hence it is only applied in the
package when the character table one works with possesses an underlying group.

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 $u \in \mathrm{V}(\mathbb{Z}G)$ of order $k$ is conjugate in $\mathbb{Q}H$, for $H$ some group containing $G$, to a trivial
unit if and only if the sum of the coefficients of $u$ at elements of order $k$ equals $1$ and the sum of coefficients of elements
of order $m$ equals $0$ for any $m \neq k$ <Cite Key="MargolisDelRioPAP"/>, Proposition 2.1. This shows that the Kimmerle Problem is in fact equvivalent to an earlier
question of A. Bovdi and hence results on Bovdi's Problem can also be applied.

For more details on when the variable <K>HeLP_sol</K> is modified or reset and how to influence this behavior see Section <Ref Sect='Chapter_Extended_examples_Section_The_behavior_of_the_variable_HeLP_sol'/> and <Ref Func='HeLP_ChangeCharKeepSols'/>.

@Section The HeLP equations

Denote by $x^G$ the conjugacy class of an element $x$ in $G$.
Let $u$ be a torsion unit in $\mathrm{V}(\mathbb{Z}G)$ of order $k$ and $D$ an ordinary
representation of $G$ over a field contained in $\mathbb{C}$ with character $\chi$.
Then $D(u)$ is a matrix of finite order and thus diagonalizable over $\mathbb{C}$.
Let $\zeta$ be a primitive $k$-th root of unity, then the multiplicity $\mu_l(u,\chi)$
of $\zeta^l$ as an eigenvalue of $D(u)$ can be computed via Fourier inversion and equals
$$ \mu_l(u,\chi) = \frac{1}{k} \sum_{1 \not= d \mid k} {\rm{Tr}}_{\mathbb{Q}(\zeta^d)/\mathbb{Q}}(\chi(u^d)\zeta^{-dl}) + \frac{1}{k} \sum_{x^G} \varepsilon_x(u) {\rm{Tr}}_{\mathbb{Q}(\zeta)/\mathbb{Q}}(\chi(x)\zeta^{-l}).$$
As this multiplicity is a non-negative integer, we have the constraints $$\mu_l(u,\chi) \in \mathbb{Z}_{\geq 0}$$
for all ordinary characters $\chi$ and all $l$.
This formula was given by I.S. Luthar and I.B.S. Passi <Cite Key="LP"/>. 
Later M. Hertweck showed that it may also be used for a representation over a field of characteristic $p > 0$
with Brauer character $\varphi$,
if $p$ is coprime to $k$ <Cite Key="HertweckBrauer"/>, § 4. In that case one has to ignore the $p$-singular conjugacy classes (i.e. the classes of elements with an order divisible by $p$) and the above formula becomes
 $$ \mu_l(u,\varphi) = \frac{1}{k} \sum_{1 \not= d \mid k} {\rm{Tr}}_{\mathbb{Q}(\zeta^d)/\mathbb{Q}}(\varphi(u^d)\zeta^{-dl}) + \frac{1}{k} \sum_{x^G,\ p \nmid o(x)} \varepsilon_x(u) {\rm{Tr}}_{\mathbb{Q}(\zeta)/\mathbb{Q}}(\varphi(x)\zeta^{-l}).$$
Again, as this multiplicity is a non-negative integer, we have the constraints $$\mu_l(u,\varphi) \in \mathbb{Z}_{\geq 0}$$
for all Brauer characters $\varphi$ and all $l$.

These equations allow to build a system of integral inequalities for the partial augmentations of $u$.
Solving these inequalities is exactly what the HeLP method does to obtain restrictions on the possible
values of the partial augmentations of $u$. Note that some of the $\varepsilon_x(u)$ are a priori zero by the results in the above sections. 
For $p$-solvable groups representations over fields of characteristic $p$ can not give any
new information compared to ordinary representations by the Fong-Swan-Rukolaine Theorem <Cite Key="CR1"/>, Theorem 22.1.

@Section The Wagner test

We also included a result motivated by a theorem R. Wagner proved 1995 in his Diplomarbeit <Cite Key="Wa"/>.
This result gives a further restriction on the partial augmentations
of torsion units. Though the results was actually available before Wagner's work, cf. <Cite Key="BovdiHertweck"/> 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 <Ref Func='HeLP_ZC'/>, <Ref Func='HeLP_PQ'/>, <Ref Func='HeLP_SP'/>, <Ref Func='HeLP_KP'/>
<Ref Func='HeLP_AllOrders'/>, <Ref Func='HeLP_AllOrdersPQ'/> and <Ref Func='HeLP_WagnerTest'/> and call it "Wagner test". 
**Theorem:** For a torsion unit <M>u \in \mathrm{V}(\mathbb{Z}G)</M>, a group element $s$,
a prime $p$ and a natural number $j$ we have
$$ \sum\limits_{x^{p^j} \sim s} \varepsilon_x(u) \equiv \varepsilon_s(u^{p^j}) \ \ \ {\rm{mod}} \ \ p. $$
Combining the Theorem with the HeLP-method may only give new insight, if $p^j$ is a proper divisor of the order of $u$.
Wagner did obtain this result for $s = 1$, when $\varepsilon_s(u) = 0$ by the Berman-Higman Theorem.
In the case that $u$ is of prime power order this is a result of J.A. Cohn and D. Livingstone <Cite Key="CohnLivingstone"/>.

@Section s-constant characters

If one is interested in units of mixed order $s*t$ for two primes $s$ and $t$ (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 <Cite Key="BKHS"/>, page 4. Assume one has several conjugacy classes of elements of order $s$, 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 $s$. 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 $s$ 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 $s$. Characters having the above property are called $s$-constant. In this way the existence of elements of order $s*t$ can be excluded in a quite efficient way without doing calculations for elements of order $s$. 
There is also the concept of $(s,t)$-constant characters, being constant on both, the conjugacy classes of elements of order $s$ and on the conjugacy classes of elements of order $t$. The implementation of this is however not yet part of this package.

@Section Known results about the Zassenhaus Conjecture and the Prime Graph Question

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: 
For the Zassenhaus Conjecture only the following reduction is available: 
**Theorem:** Assume the Zassenhaus Conjecture holds for a group $G$. Then (ZC) holds for $G \times C_2$ <Cite Key="HoefertKimmerle"/>, Corollary 3.3, and $G \times \Pi$, where $\Pi$ denotes a nilpotent group of order prime to the order of $G$ <Cite Key="HertweckEdinb"/>, Proposition 8.1. It is also known to go over to other types of direct products under certain conditions <Cite Key="BachleKimmerleSerrano"/>.
With this reductions in mind the Zassenhaus Conjecture is known for:
* Nilpotent groups <Cite Key="Weiss91"/>,
* Cyclic-By-Abelian groups <Cite Key="CyclicByAbelian"/> and some other special cyclic-by-nilpotent groups <Cite Key="CaicedoDelRio20"/>,
* Groups containing a normal Sylow subgroup with abelian complement <Cite Key="HertweckColloq"/>,
* Frobenius groups whose order is divisible by at most two different primes <Cite Key="JuriaansMilies"/>,
* Groups $X \rtimes A$, where $X$ and $A$ are abelian and $A$ is of prime order $p$ such that $p$ is smaller then any prime divisor of the order of $X$ <Cite Key="MRSW"/>,
* All groups of order up to 143 <Cite Key="BaHeKoMaSi"/>,
* The non-abelian simple groups $A_5$ <Cite Key="LP"/>, $A_6 \simeq PSL(2,9)$ <Cite Key="HerA6"/>, $PSL(2,7)$, $PSL(2,11)$, $PSL(2,13)$ <Cite Key="HertweckBrauer"/>, $PSL(2,8)$, $PSL(2,17)$ <Cite Key="KonovalovKimmiStAndrews"/> <Cite Key="Gildea"/>, $PSL(2,19)$, $PSL(2,23)$ <Cite Key="BaMaM10"/>, $PSL(2,25)$, $PSL(2,31)$, $PSL(2,32)$ <Cite Key="BaMa4prII"/> and some extensions of these groups. Also for all $PSL(2,p)$ where $p$ is a fermat or a Mersenne prime <Cite Key="FermatMersenne"/>, and $PSL(2,p)$ and $PSL(2,p^2)$ if $p \pm 1$ or $p^2 \pm 1$ is 4 multiplied by a prime <Cite Key="EiseleMargolisDefect1"/>,
* For special linear groups $SL(2,p)$ and $SL(2,p^2)$ for $p$ a prime <Cite Key="delRioSerrano19"/>.

The only known counterexamples to the conjecture are exhibited in <Cite Key="EiMa18"/>.

For the Prime Graph Question the following strong reduction was obtained in <Cite Key="KonovalovKimmiStAndrews"/>: 
**Theorem:** Assume the Prime Graph Question holds for all almost simple images of a group $G$. Then (PQ) also holds for $G.$
Here a group $G$ is called almost simple, if it is sandwiched between the inner automorphism group and the whole automorphism group of a non-abelian simple group $S$. I.e. $Inn(S) \leq G \leq Aut(S).$ Keeping this reduction in mind (PQ) is known for:
* Solvable groups <Cite Key="KimmiPQ"/>,
* All but two of the sporadic simple groups and their automorphism groups <Cite Key="CaMaBrauerTree"/>, the exceptions being the Monster and the O'Nan group; for an overview of early HeLP-results see <Cite Key="KonovalovKimmiStAndrews"/>,
* Groups whose socle is isomorphic to a group $PSL(2,p)$ or $PSL(2,p^2)$, where $p$ denotes a prime, <Cite Key="HertweckBrauer"/>, <Cite Key="BaMa4prI"/>.
* Groups whose socle is isomorphic to an alternating group, <Cite Key="SalimA7A8"/> <Cite Key="SalimA9A10"/><Cite Key="BaCa"/><Cite Key="BaMaAn"/>,
* Almost simple groups whose order is divisible by at most three different primes <Cite Key="KonovalovKimmiStAndrews"/> and <Cite Key="BaMaM10"/>. (This implies that it holds for all groups with an order divisible by at most three primes, using the reduction result above.)
* Many almost simple groups whose order is divisible by four different primes <Cite Key="BaMa4prI"/><Cite Key="BaMa4prII"/>,
* Certain infinite series of simple groups of Lie type of small rank and other groups from the character table library <Cite Key="CaMaBrauerTree"/>

Quelle purpose.autodoc Sprache: unbekannt

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