Structure of solvable rational groups (original) (raw)

2005, Proceedings of the London Mathematical Society

The restrictions on K are only seemingly very strong; there are many groups satisfying them. One hopes that there is a way of simplifying these restrictions. There is a generalisation of this result to any solvable rational group; see Theorem 4.5. However, it is worth noting that in all known examples jG=O 5 0 ðGÞj is not divisible by 3, and hence Theorem 1.2 applies. Note also that from Theorem 1.1 it follows that the Sylow 2-subgroup of a rational f2; 5g-group is rational too. This curious fact is conjectured for all rational groups in the literature, with hardly any results in this direction. With some modications to the proof of Proposition 3.1, a similar result can be achieved for nite groups in which every element is conjugate to its cube. Relevant material can be found in [5]. The philosophy of the proof is the following. One uses induction on the order of the group (x 5). With a little eort it is possible to reduce (x 5.1) to the situation where the Fitting subgroup F ðGÞ is a Sylow 5-subgroup, P , and N ¼ ZðP Þ ¼ ÈðP Þ ¼ P 0 is a minimal normal subgroup of G. The aim is to show that N ¼ ½P =N; P =N ¼ 1 (x 5.2). To this end one needs detailed information about the action of G=P on P =N (x 4). In fact, G embeds into the direct sum of wreath products with fairly small, known base groups (x 4.9). Hence P =N can be written as a direct sum of corresponding subspaces X 1 =N È. .. È X n =N. Then distinct (but similar) arguments show that ½X i ; X i ¼ 1 for any i (Proposition 5.3) and ½X i ; X j ¼ 1 for any i 6 ¼ j (Proposition 5.4). The bulk of the proof consists of exhibiting and analysing the possible primitive groups and actions, always bearing in mind what is needed for the proof of the inductive step (Theorem 4.5). The more detailed Theorem 1.2 is proven in x 3. Acknowledgements. This work was incorporated into my PhD dissertation [6]. My supervisor, Jan Saxl, contributed to its development in many ways. I would like to express my gratitude to him here. 2. Preliminary results DEFINITION 2.1. Let G act on an F-space V. The action has the eigenvector property if for any v 2 V and 2 F Ã there exists g 2 G such that vg ¼ v. The next observation of Gow lies right at the heart of the argument. PROPOSITION 2.2 [4, Lemma 4]. If G acts irreducibly on an F-space V such that every g 2 G is real in G (that is, conjugate to its inverse) then G preserves a non-singular bilinear form on V. Furthermore, if the action has the eigenvector property then jF j ¼ 2; 3; 4; 8 or the form is symplectic. The following proposition is distilled from [4, Proof of Lemma 6] or alternatively from [5, A All t a as 22 and 24]. PROPOSITION 2.3. Let G be a nite rational f2; 5g-group, P 2 Syl 5 ðGÞ, and S 2 Syl 2 ðGÞ. Suppose P is normal and elementary Abelian, so write P additively. Then S acts on P linearly. Let T be any maximal subgroup in the set fC S ðvÞ j 0 6 ¼ v 2 P g and put W ¼ C P ðT Þ and N ¼ N S ðW Þ.