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2 The main functions
 2.1 Zassenhaus Conjecture
 2.2 Prime Graph Question

2 The main functions

2.1 Zassenhaus Conjecture

This function checks whether the Zassenhaus Conjecture ((ZC) for short, cf. Section 5.1) can be proved using the HeLP method with the data available in GAP.

2.1-1 HeLP_ZC
‣ HeLP_ZC( OrdinaryCharacterTable|Group )( function )

Returns: true if (ZC) can be solved using the given data, false otherwise

HeLP_ZC 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 true without performing any calculations.

If 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 \(p\) for which the group is not \(p\)-solvable, all \(p\)-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 5.4. In case the information suffices to obtain a proof for the Zassenhaus Conjecture for this group the function will return true and false otherwise. The possible partial augmentations for elements of order \(k\) and all its powers will also be stored in the list entry HeLP_sol[k].

The prior computed partial augmentations in HeLP_sol will not be used and will be overwritten. If you do not like the last fact, please use HeLP_AllOrders (3.3-1).

2.2 Prime Graph Question

This function checks whether the Prime Graph Question ((PQ) for short, cf. Section 5.1) can be verified using the HeLP method with the data available in GAP.

2.2-1 HeLP_PQ
‣ HeLP_PQ( OrdinaryCharacterTable|Group )( function )

Returns: true if (PQ) can be solved using the given data, false otherwise

HeLP_PQ 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 true without performing any calculations.

If the group is non-solvable, the ordinary character table and all \(p\)-Brauer tables for primes \(p\) for which the group is not \(p\)-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 5.4. In case the information suffices to obtain an affirmative answer for the Prime Graph Question, the function will return true and it will return false otherwise. Let \(p\) and \(q\) be distinct primes such that there are elements of order \(p\) and \(q\) in \(G\) but no elements of order \(pq\). Then for \(k\) being \(p\), \(q\) or \(pq\) the function will save the possible partial augmentations for elements of order \(k\) and its (non-trivial) powers in HeLP_sol[k]. The function also does not use the previously computed partial augmentations for elements of these orders but will overwrite the content of HeLP_sol. If you do not like the last fact, please use HeLP_AllOrdersPQ (3.3-2).

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