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 [BM15].

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 long standing conjecture of H.J. Zassenhaus asserts 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 nowadays still open, 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.

Considering the difficulty of the problem 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 \((p, q)\).

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 \(G.\)

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.

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_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 A. 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.

With this reductions in mind the Zassenhaus Conjecture is known for:

Nilpotent groups [Wei91],

Cyclic-By-Abelian groups [CMR13],

Groups containing a normal Sylow subgroup with abelian complement [Her06],

Frobenius groups whose order is divisible by at most two different primes [JM00],

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 [BHKMS16],

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], \(PSL(2,19)\), \(PSL(2,23)\) [BM14], \(PSL(2,25)\), \(PSL(2,31)\), \(PSL(2,32)\) [BM16b] and some extensions of these groups. Also for all \(PSL(2,p)\) where \(p\) is a fermat or a Mersenne prime [MRS16].

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],

Groups whose socle is isomorphic to a group \(PSL(2,p)\) or \(PSL(2,p^2)\), where \(p\) denotes a prime, [Her07], [BM16a].

Half of the sporadic simple groups and their automorphism groups; for an overview see [KK15],

Groups whose socle is isomorphic to an alternating group of degree at most 17, [Sal11] [Sal13][BC15],

Almost simple groups whose order is divisible by at most three different primes [KK15] and [BM14]. (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 [BM16a][BM16b].

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