On the Determination of Sylow Numbers

Abstract

Let G be a finite group and denote by ${\mathbb {Z}}G$ its integral group ring. In this note we study whether groups with isomorphic integral group ring have the same Sylow numbers. We show that the Sylow q - numbers (i.e. the number of Sylow q - subgroups) coincide provided ${\mathbb {Z}}G \cong {\mathbb {Z}}H$ and G is q-constrained. If additionally $O_{q'}(G)$ is soluble the set $\mathrm{sn}(G)$ of all Sylow numbers of G is determined by ${\mathbb {Z}}G.$ This holds as well in the cases when G has dihedral Sylow 2-subgroups or when all Sylow subgroups are abelian. G. Navarro raised the question whether even the ordinary character table $\mathrm{X}(G)$ of a finite group G determines the Sylow numbers of G. We prove that this is the case when G is nilpotent-by-nilpotent, quasinilpotent, a Frobenius group or a 2-Frobenius group. In particular Sylow numbers of supersoluble groups are given by their ordinary character table. It is proved that for a finite group G with more than two prime graph components ${\mathbb {Z}}G$ determines $\mathrm{sn}(G).$ For this we show that almost simple groups of several series of finite groups of Lie type are characterized by their ordinary character table up to isomorphism. An essential tool for many results is that the Sylow numbers of a finite group G are group - theoretically determined from those of factor groups provided G has more than one minimal normal subgroup. In an appendix a short survey on Sylow like theorems for integral group rings is given by the first author. Through all sections several open problems are stated.

Appendix: On Sylow like theorems in integral group rings

by W. Kimmerle

Let G be a finite group. Denote its integral group ring by ${\mathbb {Z}}G$ and let $V({\mathbb {Z}}G)$ be the group of normalized units of ${\mathbb {Z}}G$ (i.e. units with augmentation 1). For the background on units of integral group rings see [32, 33] or [60].

The question in which classes of infinite groups Sylow’s theorem or a variant of it holds is an old topic, cf. [11] or [13]. Note that for a given prime p a Sylow p - subgroup of an infinite group H is a maximal p - subgroup. By Zorn’s lemma each p - subgroup of H is contained in a Sylow p - subgroup.

Sylow p - subgroups of countable groups may be non-isomorphic, for examples see [46]. The unit group $U({\mathbb {Z}}G)$ of an integral group ring of a finite group is a countable group. The few exceptional cases when this unit group is finite have been completely classified by G.Higman [27]. He showed also that, if G is finite abelian, then torsion units of $V({\mathbb {Z}}G)$ are trivial units, i.e they are of the form $\pm g \in G .$ It follows that Sylow’s theorem is valid for integral group rings of abelian groups (even for infinite abelian groups). However for the smallest non-abelian group $S_3$ there are involutions in $V({\mathbb {Z}}S_3)$ which are not conjugate to involutions of $S_3 .$ These involutions are conjugate in ${\mathbb {Q}}S_3$ to an involution of $S_3$, see e.g. [37]. Note that this conjugation is not an automorphism of ${\mathbb {Z}}G.$ As the research of the last decades has shown the following question is nowadays in the centre of the study of torsion units of integral group rings.

Main problem. Let G be a finite group, let p be a prime and U be a finite p - subgroup of $V({\mathbb {Z}}G).$ Is U conjugate by a unit of ${\mathbb {Q}}G$ to a subgroup of G?

If this is the case then we say that p-ZC3 holds for ${\mathbb {Z}}G.$ The abbreviation is justified because its content is precisely the third Zassenhaus ZC3 conjecture for p - subgroups. Note that for all three Zassenhaus conjectures counterexamples have been constructed, for a recent survey we recommend [52]. A. Weiss has shown that p-ZC3 holds for ${\mathbb {Z}}G$ provided G has a normal p- subgroup [60, 41.12]. It is also valid if G is p - constrained and Sylow p - subgroups are abelian [5, Proposition 3.2]. It is open whether p-ZC3 holds for each prime p and each finite group G.

If p-ZC3 is valid for a given finite group G and all primes p we say that a Sylowlike theorem holds for ${\mathbb {Z}}G.$⁴

Problem A. Find, as a first step towards the main problem, interesting classes of finite groups such that in ${\mathbb {Z}}G$ a Sylowlike theorem holds for each group G of this class.

Problem A is solved for nilpotent-by-nilpotent groups [20] and for Frobenius groups [20, 21, 35, 15, 40]. It is especially open when G is soluble. Other interesting classes of finite groups are especially those ones which played a special role for integral group rings in related areas. This includes for instance groups whose integral group rings have only trivial units, see [4, 10] or groups whose rational group algebras may be described with the aid of Shoda pairs, see e.g. [9, 34].

Note that $V({\mathbb {Z}}G)$ is a subgroup of $\mathrm{GL}(|G|,{\mathbb {Z}}).$ So evidence that p-ZC3 holds (at least for many classes of finite groups) is given by the result of H.Abold and W.Plesken that p - subgroups of $\mathrm{GL}(n,{\mathbb {Z}})$ are conjugate in $\mathrm{GL}(n,{\mathbb {Q}})$ to a subgroup of a Sylow p - subgroup of $\mathrm{GL}(n,{\mathbb {Z}})$ [1].

On the other hand in the 2 - adic group ring ${\hat{{\mathbb {Z}}}}_2 S_4$ there are two dihedral groups of order 8 which are not conjugate within ${\hat{{\mathbb {Q}}}}_2S_4$ [58, p.141, §2].

It is well known that each torsion subgroup U of $V({\mathbb {Z}}G)$ is finite and that |U| divides |G|. Thus p-ZC3 is a question only for $p \in \pi (G).$ Because the main problem may have for some groups a negative answer it is of interest to study (at least first) weaker properties. If for each prime p each p - subgroup U of $V({\mathbb {Z}}G)$ is isomorphic to a subgroup of G then we say that a weak Sylowlike theorem holds for ${\mathbb {Z}}G.$ This terminology has been introduced in [40].

Very little is known for insoluble groups. Taking off with the article of I.Luthar and I.B.S. Passi [47] character theoretical methods and algorithms have been developed to attack such groups, for an overview see [6]. In the mean time a weak Sylowlike theorem has been achieved for the series $\mathrm{PSL}(2,r^f)$ in the case when $f \le 3$ [7, 24]. Moreover p-ZC3 is known for these groups if $p \ne r$ or $p = r = 2$ or $f = 1.$

For groups G and H the isomorphism problem IP is the question whether ${\mathbb {Z}}G \cong {\mathbb {Z}}H$ implies $G \cong H .$ Suppose that G is finite. Then a subgroup H of $V({\mathbb {Z}}G)$ of the same order as G consists of ${\mathbb {Z}}$ - linearly independent elements and ${\mathbb {Z}}G \cong {\mathbb {Z}}H$ follows. Such a subgroup is called a group basis of ${\mathbb {Z}}G.$ Moreover, if ${\mathbb {Z}}G \cong {\mathbb {Z}}H$, then H is isomorphic to a group basis of G. The example of ${\mathbb {Z}}S_3$ mentioned above shows that finite subgroups of the units with augmentation 1 are in general not contained in a group basis. Because in general IP has a negative answer [23], subgroups contained in a group basis play a special role. The question arises how much group bases may differ from each other. It is natural to ask whether Sylow p-subgroups of group bases G and H are isomorphic or even conjugate within ${\mathbb {Q}}G.$ Clearly this question is a special case of the main problem. But its relation to IP shows that it deserves extra attention. Moreover in this situation it makes sense to consider as well the numbers of Sylow p - subgroups and to pose the question whether different group bases have the same Sylow numbers.

We say that a Sylowlike theorem holds for group bases of ${\mathbb {Z}}G$ provided ${\mathbb {Z}}G \cong {\mathbb {Z}}H$ implies that Sylow p - subgroups of G and H are conjugate within ${\mathbb {Q}}G$ and $\mathrm{sn}(G)$ and $\mathrm{sn}(H)$ conincide.

Problem B. Classify the classes of finite groups such that a Sylowlike theorem is valid for group bases of ${\mathbb {Z}}G$ for each group of this class.

If G is soluble then a Sylowlike theorem for group bases of ${\mathbb {Z}}G$ holds. If G is p - constrained then Sylow p - subgroups of different group bases of ${\mathbb {Z}}G$ are conjugate within ${\mathbb {Q}}G.$ This follows from [41] and Theorem 3.6.

If in ${\mathbb {Z}}G$ the second Zassenhaus conjecture holds, i.e. group bases are conjugate by a unit of ${\mathbb {Q}}G$, then Sylow subgroups of group bases are conjugate in ${\mathbb {Q}}G$. Of course in this situation the Sylow numbers coincide and thus a Sylowlike theorem for group bases holds.

In integral group rings elements of group bases are rationally conjugate because they are in class sum correspondence, see 3.1 (ii). Suppose that Sylow p - subgroups of different group bases of ${\mathbb {Z}}G$ are isomorphic. Let H be a group base of ${\mathbb {Z}}G$ and let $\tau$ be a group isomorphism from $P_G$ to $P_H ,$ where $P_G, P_H$ rsp. denote Sylow p - subgroups of G, H rsp. By [51], see also [52, Proposition 5.1] we may embed G into a larger group E such that in E elements are conjugate if, and only if, they have the same order. This will always work for $E = S_G$, where G acts on $S_G$, the symmetric group on G, just via group multiplication. Thus $\tau (g)$ is conjugate in ${\mathbb {Q}}E$ to g for each $g \in P_G .$ It follows that $P_G$ and $P_H$ have in each Wedderburn block of ${\mathbb {Q}}E$ the same characters and therefore they are conjugate in ${\mathbb {Q}}E.$ In other words enlarging the group leads to conjugation in a larger group ring.

Similar arguments may be applied in the situation when p - elements of $V({\mathbb {Z}}G)$ are rationally conjugate to elements of G, i.e. the first Zassenhaus conjecture holds for such elements. It follows that isomorphic p - subgroups of $V({\mathbb {Z}}G)$ are rationally conjugate in a group ring of a larger group. Note in this context that the known counterexamples to the first Zassenhaus conjecture [22] involve only torsion units whose order is divisible by at least two different primes.

So, if the first Zassenhaus conjecture (for p - elements) and a weak Sylowlike theorem are valid for ${\mathbb {Z}}G$, then each finite p - subgroup of $V({\mathbb {Z}}G)$ is conjugate in ${\mathbb {Q}}S_G$ to a subgroup of G. This variant of Sylowlike results is in the same spirit as replacements for the first Zassenhaus conjecture concerning arbitrary torsion units which have been studied in [8].