Elementary abelian subgroups: from algebraic groups to finite groups (Q6631305)

The paper studies the elementary abelian \(p\)-subgroups of simple linear algebraic groups and their corresponding finite groups of Lie type. When considering a linear algebraic group, the authors always work over an algebraically closed field \(K\) of characteristic \(\ell\) such that \(\ell \neq p\), noting that when the characteristic is equal to \(p\) analogues of their results are known from Borel-Tits theory. As a main result, the paper provides an algorithm, which the authors have implemented in Magma, for classifying toral elementary abelian \(p\)-subgroups, i.e., those contained in some maximal torus, up to conjugacy in a simple linear algebraic group of any type. Furthermore, they also calculate a classification of non-toral such subgroups for linear algebraic groups of exceptional type, thus providing a complete classification in these cases. They also discuss the structure of the normalizers and centralizers for all such subgroups. As explained in their introduction, this classification is done with an eye towards studying \(p\)-radical and \(p\)-local subgroups of finite simple groups.\N\NThe main techniques of the paper are two translations of information. Primarily, the authors apply the Lang-Steinberg theorem, which allows them to translate information about the conjugacy of subgroups from the linear algebraic group to a finite group of fixed points and vice versa. In particular, the authors work with the fixed points of a Steinberg endomorphism \(F \colon G \to G\) of their linear algebraic group, which by definition is an automorphism such that for some \(m\), \(F^m\) is a Frobenius morphism induced by the \(q\)-power map \(K \to K\) for some \(\mathbb{F}_q \subset K\). They then translate information between \(G\) and \(G^F\). Additionally, the authors employ a result of Larsen (with previous variations by other authors) which implies that the classification of elementary abelian \(p\)-subgroups is independent of \(\ell\) so long as \(\ell \neq p\). Further, it shows that when \(\ell \neq p\), the classification in \(G=G(K)\) also corresponds with the classification of the corresponding complex Lie group \(G(\mathbb{C})\). The result of Larsen is reviewed in the preliminaries and the application of Lang-Steinberg appears in Section 4.\N\NIn Section 5, the authors explain the mechanism behind their Magma implementation which classifies toral elementary abelian p-subgroups. Conjugates in \(G\) of such a subgroup contained in a maximal torus \(T\) will in fact be conjugate by an element of the normalizer \(N_G(T)\), and thus by an element of the Weyl group \(W=N_G(T)/T\). They show that one can replace \(T\) and \(W\) by the fixed points \(T^F\) for some Steinberg endomorphism and the extended Weyl group \(\tilde{W}\) (if \(G\) is not simply connected they define this to be an analogue of the extended Weyl group introduced by Tits) respectively, both of which are finite, and their Magma implementation computes conjugacy in \(T^F\) with respect to \(\tilde{W}\). This provides a classification of toral elementary abelian \(p\)-subgroups of \(G\) up to \(G\)-conjugacy. They then translate these computations to classify such subgroups of \(G^F\) up to \(G^F\)-conjugacy. In particular, the set of classes of toral elementary abelian p-subgroups in \(G^F\) will be the set of such classes in \(G\) which have a representative which is already a subgroup of \(G^F\). They also describe how to determine the normalizer and centralizer structure of the resulting class in \(G^F\) using the data from \(G\).\N\NThe final section is about non-toral elementary abelian subgroups in exceptional algebraic groups. In this case, conjugates of such groups are no longer necessarily conjugate in the normalizer of a torus, and so reasoning with the Weyl group is not applicable. Nevertheless, the transfer techniques from characteristic zero are still applicable. Because of this, the authors are able to amalgamate previous calculations from the literature, together with some of their own calculations for the adjoint group of type \(E_7\) in the case \(p=2\), and they present the classification of these subgroups in detailed tables. The arguments in this section are rather technical and the authors point out that they are at times ad-hoc. Some additional supporting details for the calculations presented are included in the two appendices.