  
  [1X2 [33X[0;0YThe main functions[133X[101X
  
  
  [1X2.1 [33X[0;0YZassenhaus Conjecture[133X[101X
  
  [33X[0;0YThis  function checks whether the Zassenhaus Conjecture ((ZC) for short, cf.
  Section  [14X5.1[114X) can be proved using the HeLP method with the data available in
  GAP.[133X
  
  [1X2.1-1 HeLP_ZC[101X
  
  [33X[1;0Y[29X[2XHeLP_ZC[102X( [3XOrdinaryCharacterTable|Group[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if (ZC) can be solved using the given data, [9Xfalse[109X otherwise[133X
  
  [33X[0;0Y[9XHeLP_ZC[109X checks whether the Zassenhaus Conjecture can be solved for the given
  group  using  the  HeLP  method,  the  Wagner  test  and  all character data
  available.  The argument of the function can be either an ordinary character
  table  or  a  group.  In  the  second  case  it  will  first  calculate  the
  corresponding  ordinary  character  table.  If  the  group  in  question  is
  nilpotent,  the  Zassenhaus Conjecture holds by a result of A. Weiss and the
  function will return [9Xtrue[109X without performing any calculations.[133X
  
  [33X[0;0YIf  the  group is not solvable, the function will check all orders which are
  divisors  of  the  exponent  of the group. If the group is solvable, it will
  only check the orders of group elements, as there can't be any torsion units
  of  another  order.  The  function  will use the ordinary table and, for the
  primes  [23Xp[123X  for  which the group is not [23Xp[123X-solvable, all [23Xp[123X-Brauer tables which
  are  available in GAP to produce as many constraints on the torsion units as
  possible.  Additionally,  the  Wagner  test  is  applied to the results, cf.
  Section  [14X5.4[114X.  In  case  the  information suffices to obtain a proof for the
  Zassenhaus Conjecture for this group the function will return [9Xtrue[109X and [9Xfalse[109X
  otherwise.  The  possible  partial augmentations for elements of order [22Xk[122X and
  all its powers will also be stored in the list entry [9XHeLP_sol[k][109X.[133X
  
  [33X[0;0YThe  prior  computed  partial augmentations in [9XHeLP_sol[109X will not be used and
  will  be  overwritten.  If  you  do  not  like  the  last  fact,  please use
  [2XHeLP_AllOrders[102X ([14X3.3-1[114X).[133X
  
  
  [1X2.2 [33X[0;0YPrime Graph Question[133X[101X
  
  [33X[0;0YThis  function  checks whether the Prime Graph Question ((PQ) for short, cf.
  Section  [14X5.1[114X)  can be verified using the HeLP method with the data available
  in GAP.[133X
  
  [1X2.2-1 HeLP_PQ[101X
  
  [33X[1;0Y[29X[2XHeLP_PQ[102X( [3XOrdinaryCharacterTable|Group[103X ) [32X function[133X
  [6XReturns:[106X  [33X[0;10Y[9Xtrue[109X if (PQ) can be solved using the given data, [9Xfalse[109X otherwise[133X
  
  [33X[0;0Y[9XHeLP_PQ[109X  checks  whether  an affirmative answer for the Prime Graph Question
  for  the  given  group  can  be  obtained  using the HeLP method, the Wagner
  restrictions  and  the  data  available. The argument of the function can be
  either  an  ordinary  character table or a group. In the second case it will
  first  calculate the corresponding ordinary character table. If the group in
  question  is solvable, the Prime Graph Question has an affirmative answer by
  a result of W. Kimmerle and the function will return [9Xtrue[109X without performing
  any calculations.[133X
  
  [33X[0;0YIf  the group is non-solvable, the ordinary character table and all [23Xp[123X-Brauer
  tables  for  primes  [23Xp[123X  for  which the group is not [23Xp[123X-solvable and which are
  available  in GAP will be used to produce as many constraints on the torsion
  units  as possible. Additionally, the Wagner test is applied to the results,
  cf.  Section  [14X5.4[114X. In case the information suffices to obtain an affirmative
  answer  for  the  Prime Graph Question, the function will return [9Xtrue[109X and it
  will  return [9Xfalse[109X otherwise. Let [23Xp[123X and [23Xq[123X be distinct primes such that there
  are  elements  of order [23Xp[123X and [23Xq[123X in [23XG[123X but no elements of order [23Xpq[123X. Then for [23Xk[123X
  being  [23Xp[123X,  [23Xq[123X or [23Xpq[123X the function will save the possible partial augmentations
  for  elements  of  order  [23Xk[123X and its (non-trivial) powers in [9XHeLP_sol[k][109X. The
  function also does not use the previously computed partial augmentations for
  elements  of these orders but will overwrite the content of [9XHeLP_sol[109X. If you
  do not like the last fact, please use [2XHeLP_AllOrdersPQ[102X ([14X3.3-2[114X).[133X
  
