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 [BM18].
Let \(G\) be a finite group and let \(\mathbb{Z}G\) denote its integral group ring. Let \(\mathrm{V}(\mathbb{Z}G)\) be the group of units of augmentation one, aka. normalized units. An element of the unit group of \(\mathbb{Z}G\) is called a torsion element, if it has finite order.
A conjecture of H.J. Zassenhaus asserted that every normalized torsion unit of \(\mathbb{Z}G\) is conjugate within \(\mathbb{Q}G\) ("rationally conjugate") to an element of \(G\), 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 \(\mathbb{Z}G\) is as far determined by \(G\) as possible.
Negative solutions to the conjecture were finally found in [EM18].
Considering the difficulty of the Zassenhaus Conjecture W. Kimmerle raised the question, whether the Prime Graph of the normalized units of \(\mathbb{Z}G\) coincides with that one of \(G\) (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 \(p\), for which there is an element of order \(p\) in the group. Two vertices \(p\) and \(q\) are joined by an edge, provided there is an element of order \(pq\) in the group. In the light of this description, the Prime Graph Question asks, whether there exists an element of order \(pq\) in \(G\) provided there exists an element of order \(pq\) in \(\mathrm{V}(\mathbb{Z}G)\) 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 \(G\) and \(\mathrm{V}(\mathbb{Z}G)\) 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:
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 [Kim07] asks if any unit of finite order in \(\mathrm{V}(\mathbb{Z}G)\) is conjugate in the rational group algebra \(\mathbb{Q}H\) to a trivial unit, where \(H\) is a finite group containing \(G\). 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.
For a finite group \(G\) and an element \(x \in G\) let \(x^G\) denote the conjugacy class of \(x\) in \(G\). The partial augmentation with respect to \(x\) or rather the conjugacy class of \(x\) is the map \(\varepsilon_x \) sending an element \(u\) to the sum of the coefficients at elements of the conjugacy class of \(x\), 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 \(u\) be a torsion element in \(\mathrm{V}(\mathbb{Z}G)\). By results of G. Higman, S.D. Berman and M. Hertweck the following is known for the partial augmentations of \(u\):
Theorem: ([Seh93], Proposition (1.4); [Her07], 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 [MRSW87], Theorem 2.5:
Theorem: A torsion unit \(u \in \mathrm{V}(\mathbb{Z}G)\) of order \(k\) is rationally conjugate to an element of \(G\) if and only if all partial augmentations of \(u^d\) vanish, except one (which then is necessarily 1) for all divisors \(d\) of \(k\).
The last statement also explains the structure of the variable HeLP_sol. In HeLP_sol[k] the possible partial augmentations for an element of order \(k\) and all powers \(u^d\) for \(d\) dividing \(k\) (except for \(d=k\)) are stored, sorted ascending w.r.t. order of the element \(u^d\). For instance, for \(k = 12\) an entry of HeLP_sol[12] might be of the following form:
[ [ 1 ],[ 0, 1 ],[ -2, 2, 1 ],[ 1, -1, 1 ],[ 0, 0, 0, 1, -1, 0, 1, 0, 0 ] ].
The first sublist [ 1 ] indicates that the element \(u^6\) of order 2 has the partial augmentation 1 at the only class of elements of order 2, the second sublist [ 0, 1 ] indicates that \(u^4\) 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 [ -2, 2, 1 ] states that the element \(u^3\) 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 \(u\) 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 \(z\) is a central element in \(G\) and \(u \in \mathrm{V}(\mathbb{Z}G)\) is a torsion unit different from \(z\), then \(\varepsilon_z(u)= 0\). Moreover, two more elaborate criteria derived from the work of Hertweck are used:
Theorem:([Her08a], Proposition 2; [Her08b], Lemma 2.2; [Mar17]) Let \(u \in \mathrm{V}(\mathbb{Z}G)\) be of finite order and \(\varepsilon_g(u) \neq 0\) for some \(g \in G\). Suppose that \(u\) has smaller order modulo some normal \(p\)-subgroup \(N\) of \(G\). Then the \(p\)-part of \(g\) has the same order as the \(p\)-part of \(u\). Furthermore, if the \(p\)-part of \(u\) is \(p\)-adically conjugate to an element in \(G\), then the \(p\)-part of \(g\) is even conjugate in \(G\) to the \(p\)-part of \(u\). Such a \(p\)-adic conjugation holds, if the order of \(u\) modulo a normal \(p\)-subgroup of \(G\) is not divisible by \(p\), 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 \(G\) 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 \(m \neq k\) [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.
For more details on when the variable HeLP_sol is modified or reset and how to influence this behavior see Section 4.2 and HeLP_ChangeCharKeepSols (3.4-1).
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 [LP89].
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 \(k\) [Her07], ยง 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 [CR90], Theorem 22.1.
We 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 HeLP_ZC (2.1-1), HeLP_PQ (2.2-1), HeLP_SP (2.3-1), HeLP_KP (2.4-1) HeLP_AllOrders (3.3-1), HeLP_AllOrdersPQ (3.3-2) and HeLP_WagnerTest (3.7-1) and call it "Wagner test".
Theorem: For a torsion unit \(u \in \mathrm{V}(\mathbb{Z}G)\), 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 [CL65].
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 [BK10], 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.
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 \(G \times C_2\) [HK06], Corollary 3.3, and \(G \times \Pi\), where \(\Pi\) denotes a nilpotent group of order prime to the order of \(G\) [Her08b], Proposition 8.1.
It 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:
Nilpotent groups [Wei91],
Cyclic-By-Abelian groups [CMdR13] and some other special cyclic-by-nilpotent groups [CdR20],
Groups containing a normal Sylow subgroup with abelian complement [Her06],
Frobenius groups whose order is divisible by at most two different primes [JPM00],
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 \(X\) [MRSW87],
All groups of order up to 143 [BHK+18],
The non-abelian simple groups \(A_5\) [LP89], \(A_6 \simeq PSL(2,9)\) [Her08c], \(PSL(2,7)\), \(PSL(2,11)\), \(PSL(2,13)\) [Her07], \(PSL(2,8)\), \(PSL(2,17)\) [KK15] [Gil13], \(PSL(2,19)\), \(PSL(2,23)\) [BM17b], \(PSL(2,25)\), \(PSL(2,31)\), \(PSL(2,32)\) [BM19b] and some extensions of these groups. Also for all \(PSL(2,p)\) where \(p\) is a fermat or a Mersenne prime [MdRS19], and \(PSL(2,p)\) and \(PSL(2,p^2)\) if \(p \pm 1\) or \(p^2 \pm 1\) is 4 multiplied by a prime [EM22],
For special linear groups \(SL(2,p)\) and \(SL(2,p^2)\) for \(p\) a prime [dRS19].
The only known counterexamples to the conjecture are exhibited in [EM18].
For the Prime Graph Question the following strong reduction was obtained in [KK15]:
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 [Kim06],
All 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],
Groups whose socle is isomorphic to a group \(PSL(2,p)\) or \(PSL(2,p^2)\), where \(p\) denotes a prime, [Her07], [BM17a].
Groups whose socle is isomorphic to an alternating group, [Sal11] [Sal13][BC17][BM19a],
Almost 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.)
Many almost simple groups whose order is divisible by four different primes [BM17a][BM19b],
Certain infinite series of simple groups of Lie type of small rank and other groups from the character table library [CM21]
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