  
  [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 the Prime Graph Question[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;0YWhen  the  first versions of this package were published in 2015-2017 a long
  standing  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 was the first of his
  three  famous  conjectures about integral group rings and the only one which
  remained  open  at  the  time,  hence  it  is  referred to as the Zassenhaus
  Conjecture (ZC).[133X
  
  [33X[0;0YIn  October  2017 counterexamples to the Zassenhaus Conjecture were found by
  F. Eisele and L. Margolis [EM18].[133X
  
  [33X[0;0YConsidering  the  difficulty of the problem 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;0YIn  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
  
  
  [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;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_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 A. 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;0YA  survey  on  topics  touched  on  by this package has been written in 2018
  [MdR18].  The  counterexamples  to the Zassenhaus Conjecture are of the form
  [23X(C_{pq}  \times  C_{pq}) \rtimes A[123X where [23Xp[123X and [23Xq[123X are certain primes and [23XA[123X is
  an abelian group with a specified action [EM18].[133X
  
  [33X[0;0YIn  December  2019,  to  the  best  of our knowledge, the following positive
  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;0YWith 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],[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;6YGroups with an abelian normal subgroup of index 2 [LP92],[133X
  
  [30X    [33X[0;6YSome more special classes of metabelian groups [MR18] and [MdR19],[133X
  
  [30X    [33X[0;6YDirect  products  of  an  abelian  group  and  a  Frobenius group with
        complement of odd order [BKS18],[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],[133X
  
  [30X    [33X[0;6YFor  the  following  non-solvable  and non-simple groups: [23XS_5[123X [Her07],
        [23XGL(2,5)[123X  and  the  covering group of [23XS_5[123X [BH08] and the groups [23XSL(2,p)[123X
        and [23XSL(2,p^2)[123X for any prime [23Xp[123X [dRS19].[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;6YGroups   whose   socle   is   isomorphic   to  an  alternating  group,
        [LP89][Her08c][Sal11][Sal13][BM17b][BC17][BM19a],[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;6YA group with sporadic simple socle which is not the O'Nan group or the
        Monster
        [BK07a][BK07b][BKS07][BKL11][BKM08][BK08][BGK09][BJK11][BK12][BK09][BKL08]
        [BK10][KK15][Mar19][BM19a][CM19],[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;6YMany groups from the GAP character table library [CM19].[133X
  
