  
  [1X5 [33X[0;0YBackground[133X[101X
  
  [33X[0;0YIn  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 [BM18].[133X
  
  
  [1X5.1 [33X[0;0YThe Zassenhaus Conjecture and related questions[133X[101X
  
  [33X[0;0YLet  [23XG[123X  be  a  finite  group  and let [22XZG[122X denote its integral group ring. Let
  [22XmathrmV(ZG)[122X  be  the  group  of  units  of augmentation one, aka. normalized
  units. An element of the unit group of [22XZG[122X is called a torsion element, if it
  has finite order.[133X
  
  [33X[0;0YA  conjecture of H.J. Zassenhaus asserted that every normalized torsion unit
  of  [22XZG[122X  is  conjugate within [22XQG[122X ("rationally conjugate") to an element of [22XG[122X,
  see  [Zas74]  or  [Seh93], 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 [22XZG[122X is as far determined by [22XG[122X as possible.[133X
  
  [33X[0;0YNegative solutions to the conjecture were finally found in [EM18].[133X
  
  [33X[0;0YConsidering  the  difficulty of the Zassenhaus Conjecture W. Kimmerle raised
  the  question,  whether  the  Prime  Graph  of  the  normalized  units of [22XZG[122X
  coincides with that one of [22XG[122X (cf. [Kim07] 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 [22Xp[122X, for which there is an
  element of order [22Xp[122X in the group. Two vertices [22Xp[122X and [22Xq[122X are joined by an edge,
  provided  there is an element of order [22Xpq[122X in the group. In the light of this
  description,  the Prime Graph Question asks, whether there exists an element
  of order [23Xpq[123X in [22XG[122X provided there exists an element of order [23Xpq[123X in [22XmathrmV(ZG)[122X
  for every pair of primes [23X(p, q)[123X.[133X
  
  [33X[0;0YA  question which lies between the Zassenhaus Conjecture and the Prime Graph
  Question  is  the  Spectrum Problem. It asks, if the orders of elements in [22XG[122X
  and  [22XmathrmV(ZG)[122X  coincide.  In  general,  by  a result of J. A. Cohn and D.
  Livingstone [CL65], Corollary 4.1, and a result of M. Hertweck [Her08a], the
  following  is  known  about the possible orders of torsion units in integral
  group rings:[133X
  
  [33X[0;0Y[13XTheorem:[113X  The exponents of [23X\mathrm{V}(\mathbb{Z}G)[123X and [23XG[123X coincide. Moreover,
  if  [23XG[123X  is solvable, any torsion unit in [23X\mathrm{V}(\mathbb{Z}G)[123X has the same
  order as some element in [23XG.[123X[133X
  
  [33X[0;0YFinally,  a  question  raised  by W. Kimmerle in [Kim07] asks if any unit of
  finite order in [22XmathrmV(ZG)[122X is conjugate in the rational group algebra [22XQH[122X to
  a  trivial  unit,  where  [22XH[122X is a finite group containing [22XG[122X. 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  [EM18]  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.[133X
  
  
  [1X5.2 [33X[0;0YPartial augmentations and the structure of HeLP sol[133X[101X
  
  [33X[0;0YFor a finite group [22XG[122X and an element [22Xx ∈ G[122X let [22Xx^G[122X denote the conjugacy class
  of  [22Xx[122X  in  [22XG[122X.  The  partial  augmentation  with  respect  to [22Xx[122X or rather the
  conjugacy  class  of [22Xx[122X is the map [22Xε_x[122X sending an element [22Xu[122X to the sum of the
  coefficients at elements of the conjugacy class of [22Xx[122X, i.e.[133X
  
  
  [24X[33X[0;6Y\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.[133X
  
  [124X
  
  [33X[0;0YLet  [22Xu[122X  be  a  torsion element in [22XmathrmV(ZG)[122X. By results of G. Higman, S.D.
  Berman  and M. Hertweck the following is known for the partial augmentations
  of [23Xu[123X:[133X
  
  [33X[0;0Y[13XTheorem:[113X ([Seh93], Proposition (1.4); [Her07], Theorem 2.3) [23X\varepsilon_1(u)
  = 0[123X if [23Xu \not= 1[123X and [23X\varepsilon_x(u) = 0[123X if the order of [23Xx[123X does not divides
  the order of [23Xu[123X.[133X
  
  [33X[0;0YPartial  augmentations  are  connected  to  (ZC)  and (PQ) via the following
  result,  which  is  due  to  Z. Marciniak, J. Ritter, S. Sehgal and A. Weiss
  [MRSW87], Theorem 2.5:[133X
  
  [33X[0;0Y[13XTheorem:[113X  A  torsion unit [22Xu ∈ mathrmV(ZG)[122X of order [23Xk[123X is rationally conjugate
  to  an  element of [23XG[123X if and only if all partial augmentations of [22Xu^d[122X vanish,
  except one (which then is necessarily 1) for all divisors [22Xd[122X of [22Xk[122X.[133X
  
  [33X[0;0YThe  last statement also explains the structure of the variable [9XHeLP_sol[109X. In
  [9XHeLP_sol[k][109X the possible partial augmentations for an element of order [23Xk[123X and
  all  powers  [22Xu^d[122X  for  [22Xd[122X  dividing  [22Xk[122X  (except  for  [22Xd=k[122X) are stored, sorted
  ascending w.r.t. order of the element [22Xu^d[122X. For instance, for [22Xk = 12[122X an entry
  of [9XHeLP_sol[12][109X might be of the following form:[133X
  
  [33X[0;0Y[9X[ [ 1 ],[ 0, 1 ],[ -2, 2, 1 ],[ 1, -1, 1 ],[ 0, 0, 0, 1, -1, 0, 1, 0, 0 ] ][109X.[133X
  
  [33X[0;0YThe  first  sublist  [9X[ 1 ][109X indicates that the element [22Xu^6[122X of order 2 has the
  partial  augmentation 1 at the only class of elements of order 2, the second
  sublist [9X[ 0, 1 ][109X indicates that [22Xu^4[122X 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  [9X[  -2,  2,  1  ][109X states that the element [22Xu^3[122X 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 [22Xu[122X and its powers by the above
  restrictions on the partial augmentations.[133X
  
  [33X[0;0YFrom   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 [22Xz[122X is a
  central element in [22XG[122X and [22Xu ∈ mathrmV(ZG)[122X is a torsion unit different from [22Xz[122X,
  then  [22Xε_z(u)= 0[122X. Moreover, two more elaborate criteria derived from the work
  of Hertweck are used:[133X
  
  [33X[0;0Y[13XTheorem:[113X([Her08a],  Proposition  2;  [Her08b],  Lemma  2.2; [Mar17]) Let [22Xu ∈
  mathrmV(ZG)[122X be of finite order and [22Xε_g(u) ≠ 0[122X for some [22Xg ∈ G[122X. Suppose that [22Xu[122X
  has smaller order modulo some normal [22Xp[122X-subgroup [22XN[122X of [22XG[122X. Then the [22Xp[122X-part of [22Xg[122X
  has  the  same  order as the [22Xp[122X-part of [22Xu[122X. Furthermore, if the [22Xp[122X-part of [22Xu[122X is
  [22Xp[122X-adically  conjugate  to  an  element  in  [22XG[122X,  then the [22Xp[122X-part of [22Xg[122X is even
  conjugate  in  [22XG[122X to the [22Xp[122X-part of [22Xu[122X. Such a [22Xp[122X-adic conjugation holds, if the
  order  of  [22Xu[122X modulo a normal [22Xp[122X-subgroup of [22XG[122X is not divisible by [22Xp[122X, i.e. the
  [23Xp[123X-part of [23Xu[123X is trivial modulo a normal [23Xp[123X-subgroup.[133X
  
  [33X[0;0YTo  apply  this  theorem,  some  knowledge  on  the normal subgroups of [22XG[122X is
  necessary.  Hence it is only applied in the package when the character table
  one works with possesses an underlying group.[133X
  
  [33X[0;0YIt  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  [23Xu  \in  \mathrm{V}(\mathbb{Z}G)[123X of order [23Xk[123X is
  conjugate  in  [23X\mathbb{Q}H[123X, for [23XH[123X some group containing [23XG[123X, to a trivial unit
  if  and  only  if  the  sum  of the coefficients of [23Xu[123X at elements of order [23Xk[123X
  equals [23X1[123X and the sum of coefficients of elements of order [23Xm[123X equals [23X0[123X for any
  [23Xm  \neq  k[123X [MdR19], 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.[133X
  
  [33X[0;0YFor  more details on when the variable [9XHeLP_sol[109X is modified or reset and how
  to  influence  this  behavior  see  Section  [14X4.2[114X and [2XHeLP_ChangeCharKeepSols[102X
  ([14X3.4-1[114X).[133X
  
  
  [1X5.3 [33X[0;0YThe HeLP equations[133X[101X
  
  [33X[0;0YDenote  by  [23Xx^G[123X the conjugacy class of an element [23Xx[123X in [23XG[123X. Let [23Xu[123X be a torsion
  unit  in [23X\mathrm{V}(\mathbb{Z}G)[123X of order [23Xk[123X and [23XD[123X an ordinary representation
  of  [23XG[123X over a field contained in [23X\mathbb{C}[123X with character [23X\chi[123X. Then [23XD(u)[123X is
  a  matrix of finite order and thus diagonalizable over [23X\mathbb{C}[123X. Let [23X\zeta[123X
  be  a  primitive  [23Xk[123X-th root of unity, then the multiplicity [23X\mu_l(u,\chi)[123X of
  [23X\zeta^l[123X  as  an eigenvalue of [23XD(u)[123X can be computed via Fourier inversion and
  equals[133X
  
  
  [24X[33X[0;6Y\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}).[133X
  
  [124X
  
  [33X[0;0YAs this multiplicity is a non-negative integer, we have the constraints[133X
  
  
  [24X[33X[0;6Y\mu_l(u,\chi) \in \mathbb{Z}_{\geq 0}[133X
  
  [124X
  
  [33X[0;0Yfor  all  ordinary characters [23X\chi[123X and all [23Xl[123X. This formula was given by I.S.
  Luthar and I.B.S. Passi [LP89].[133X
  
  [33X[0;0YLater  M. Hertweck showed that it may also be used for a representation over
  a  field  of  characteristic  [23Xp  >  0[123X with Brauer character [23X\varphi[123X, if [23Xp[123X is
  coprime  to  [23Xk[123X  [Her07],  § 4. In that case one has to ignore the [23Xp[123X-singular
  conjugacy  classes  (i.e. the classes of elements with an order divisible by
  [23Xp[123X) and the above formula becomes[133X
  
  
  [24X[33X[0;6Y\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}).[133X
  
  [124X
  
  [33X[0;0YAgain,  as  this  multiplicity  is  a  non-negative  integer,  we  have  the
  constraints[133X
  
  
  [24X[33X[0;6Y\mu_l(u,\varphi) \in \mathbb{Z}_{\geq 0}[133X
  
  [124X
  
  [33X[0;0Yfor all Brauer characters [23X\varphi[123X and all [23Xl[123X.[133X
  
  [33X[0;0YThese  equations  allow  to  build a system of integral inequalities for the
  partial  augmentations  of [23Xu[123X. Solving these inequalities is exactly what the
  HeLP  method  does  to  obtain  restrictions  on  the possible values of the
  partial  augmentations  of  [23Xu[123X.  Note that some of the [23X\varepsilon_x(u)[123X are a
  priori zero by the results in the above sections.[133X
  
  [33X[0;0YFor  [23Xp[123X-solvable  groups  representations over fields of characteristic [23Xp[123X can
  not  give  any  new  information compared to ordinary representations by the
  Fong-Swan-Rukolaine Theorem [CR90], Theorem 22.1.[133X
  
  
  [1X5.4 [33X[0;0YThe Wagner test[133X[101X
  
  [33X[0;0YWe  also  included  a result motivated by a theorem R. Wagner proved 1995 in
  his  Diplomarbeit  [Wag95].  This  result gives a further restriction on the
  partial  augmentations  of  torsion  units.  Though the results was actually
  available before Wagner's work, cf. [BH08] 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 [2XHeLP_ZC[102X ([14X2.1-1[114X), [2XHeLP_PQ[102X ([14X2.2-1[114X), [2XHeLP_SP[102X
  ([14X2.3-1[114X),  [2XHeLP_KP[102X  ([14X2.4-1[114X)  [2XHeLP_AllOrders[102X ([14X3.3-1[114X), [2XHeLP_AllOrdersPQ[102X ([14X3.3-2[114X)
  and [2XHeLP_WagnerTest[102X ([14X3.7-1[114X) and call it "Wagner test".[133X
  
  [33X[0;0Y[13XTheorem:[113X  For  a  torsion unit [22Xu ∈ mathrmV(ZG)[122X, a group element [23Xs[123X, a prime [23Xp[123X
  and a natural number [23Xj[123X we have[133X
  
  
  [24X[33X[0;6Y\sum\limits_{x^{p^j}  \sim s} \varepsilon_x(u) \equiv \varepsilon_s(u^{p^j})
  \ \ \ {\rm{mod}} \ \ p.[133X
  
  [124X
  
  [33X[0;0YCombining the Theorem with the HeLP-method may only give new insight, if [23Xp^j[123X
  is a proper divisor of the order of [23Xu[123X. Wagner did obtain this result for [23Xs =
  1[123X,  when [23X\varepsilon_s(u) = 0[123X by the Berman-Higman Theorem. In the case that
  [23Xu[123X  is  of prime power order this is a result of J.A. Cohn and D. Livingstone
  [CL65].[133X
  
  
  [1X5.5 [33X[0;0Ys-constant characters[133X[101X
  
  [33X[0;0YIf  one  is  interested  in  units of mixed order [23Xs*t[123X for two primes [23Xs[123X and [23Xt[123X
  (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 [BK10],
  page 4. Assume one has several conjugacy classes of elements of order [23Xs[123X, 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 [23Xs[123X. 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 [23Xs[123X 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 [23Xs[123X. Characters having the
  above  property are called [23Xs[123X-constant. In this way the existence of elements
  of  order  [23Xs*t[123X  can  be  excluded  in  a  quite  efficient way without doing
  calculations for elements of order [23Xs[123X.[133X
  
  [33X[0;0YThere  is  also  the concept of [23X(s,t)[123X-constant characters, being constant on
  both,  the  conjugacy  classes  of  elements of order [23Xs[123X and on the conjugacy
  classes  of  elements  of order [23Xt[123X. The implementation of this is however not
  yet part of this package.[133X
  
  
  [1X5.6  [33X[0;0YKnown  results  about  the  Zassenhaus  Conjecture  and the Prime Graph[101X
  [1XQuestion[133X[101X
  
  [33X[0;0YAt  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:[133X
  
  [33X[0;0YFor the Zassenhaus Conjecture only the following reduction is available:[133X
  
  [33X[0;0Y[13XTheorem:[113X  Assume  the  Zassenhaus  Conjecture holds for a group [23XG[123X. Then (ZC)
  holds  for  [23XG  \times C_2[123X [HK06], Corollary 3.3, and [23XG \times \Pi[123X, where [23X\Pi[123X
  denotes  a  nilpotent  group  of  order  prime  to  the order of [23XG[123X [Her08b],
  Proposition 8.1.[133X
  
  [33X[0;0YIt  is also known to go over to other types of direct products under certain
  conditions  [BKS20].  With this reductions in mind the Zassenhaus Conjecture
  is known for:[133X
  
  [30X    [33X[0;6YNilpotent groups [Wei91],[133X
  
  [30X    [33X[0;6YCyclic-By-Abelian    groups    [CMdR13]   and   some   other   special
        cyclic-by-nilpotent groups [CdR20],[133X
  
  [30X    [33X[0;6YGroups  containing  a  normal  Sylow  subgroup with abelian complement
        [Her06],[133X
  
  [30X    [33X[0;6YFrobenius  groups  whose  order  is divisible by at most two different
        primes [JPM00],[133X
  
  [30X    [33X[0;6YGroups  [23XX \rtimes A[123X, where [23XX[123X and [23XA[123X are abelian and [23XA[123X is of prime order
        [23Xp[123X  such  that  [23Xp[123X  is  smaller then any prime divisor of the order of [23XX[123X
        [MRSW87],[133X
  
  [30X    [33X[0;6YAll groups of order up to 143 [BHK+18],[133X
  
  [30X    [33X[0;6YThe   non-abelian  simple  groups  [23XA_5[123X  [LP89],  [23XA_6  \simeq  PSL(2,9)[123X
        [Her08c],  [23XPSL(2,7)[123X, [23XPSL(2,11)[123X, [23XPSL(2,13)[123X [Her07], [23XPSL(2,8)[123X, [23XPSL(2,17)[123X
        [KK15]  [Gil13],  [23XPSL(2,19)[123X,  [23XPSL(2,23)[123X [BM17b], [23XPSL(2,25)[123X, [23XPSL(2,31)[123X,
        [23XPSL(2,32)[123X  [BM19b]  and  some extensions of these groups. Also for all
        [23XPSL(2,p)[123X  where  [23Xp[123X  is  a  fermat  or  a  Mersenne prime [MdRS19], and
        [23XPSL(2,p)[123X  and  [23XPSL(2,p^2)[123X if [23Xp \pm 1[123X or [23Xp^2 \pm 1[123X is 4 multiplied by a
        prime [EM22],[133X
  
  [30X    [33X[0;6YFor special linear groups [23XSL(2,p)[123X and [23XSL(2,p^2)[123X for [23Xp[123X a prime [dRS19].[133X
  
  [33X[0;0YThe only known counterexamples to the conjecture are exhibited in [EM18].[133X
  
  [33X[0;0YFor  the Prime Graph Question the following strong reduction was obtained in
  [KK15]:[133X
  
  [33X[0;0Y[13XTheorem:[113X  Assume the Prime Graph Question holds for all almost simple images
  of a group [23XG[123X. Then (PQ) also holds for [23XG.[123X[133X
  
  [33X[0;0YHere  a  group  [23XG[123X  is  called almost simple, if it is sandwiched between the
  inner  automorphism  group and the whole automorphism group of a non-abelian
  simple  group  [23XS[123X.  I.e. [23XInn(S) \leq G \leq Aut(S).[123X Keeping this reduction in
  mind (PQ) is known for:[133X
  
  [30X    [33X[0;6YSolvable groups [Kim06],[133X
  
  [30X    [33X[0;6YAll  but  two  of  the  sporadic  simple groups and their automorphism
        groups  [CM21],  the exceptions being the Monster and the O'Nan group;
        for an overview of early HeLP-results see [KK15],[133X
  
  [30X    [33X[0;6YGroups  whose  socle  is isomorphic to a group [23XPSL(2,p)[123X or [23XPSL(2,p^2)[123X,
        where [23Xp[123X denotes a prime, [Her07], [BM17a].[133X
  
  [30X    [33X[0;6YGroups  whose  socle  is  isomorphic  to an alternating group, [Sal11]
        [Sal13][BC17][BM19a],[133X
  
  [30X    [33X[0;6YAlmost  simple  groups  whose  order  is  divisible  by  at most three
        different  primes  [KK15] and [BM17b]. (This implies that it holds for
        all  groups with an order divisible by at most three primes, using the
        reduction result above.)[133X
  
  [30X    [33X[0;6YMany  almost  simple groups whose order is divisible by four different
        primes [BM17a][BM19b],[133X
  
  [30X    [33X[0;6YCertain infinite series of simple groups of Lie type of small rank and
        other groups from the character table library [CM21][133X
  
