TI subgroups and depth 3 subgroups in simple Suzuki groups

Hayder Abbas Janabi,Erzsébet Horváth,László Héthelyi

doi:10.1515/jgth-2020-0044

Hayder Abbas Janabi, Erzsébet Horváth + Show 1 more

Open Access

https://doi.org/10.1515/jgth-2020-0044

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Abstract

Abstract In this paper, we determine the TI subgroups of the simple Suzuki groups Sz ⁢ ( q ) \mathrm{Sz}(q) . More generally, we determine those nontrivial subgroups that are disjoint from some of their conjugates. It turns out that the latter are exactly those subgroups that have ordinary depth 3. The Sylow 2-subgroups of simple Suzuki groups belong to the class of so-called Suzuki 2-groups, which have been studied extensively by Higman. These results were extended later by Goldschmidt, Shaw, Shult, Gross, Wilkens and Bryukhanova. As a corollary of our investigations, we get some interesting results for the Sylow 2-subgroups of Suzuki groups, as well. We relate this to an open problem on Suzuki 2-groups, and we ask a question concerning that. We also give some characterization of Suzuki groups.

Highlights

We will show in Theorem 1.3 that, in the case of simple Suzuki groups, subgroups of ordinary depth 3 are exactly those nontrivial subgroups that have a disjoint conjugate
Sylow 2-subgroups of simple Suzuki groups Sz.q/ are isomorphic to certain A.2m C 1; Â /, where Â 2 Aut.GF .22mC1//, which is of odd order, falls into the category (i) in the previous theorem
From Theorem 5.3, it follows that, in Question 5.5, Aut.S/ acts transitively on the involutions of S, so this conjecture is a weakened form of a conjecture of Gross on finite 2-groups S with more than 1 involution admitting a group of automorphisms that transitively permutes the involutions of S, called 2-automorphic 2-groups; see [11]

Summary

Motivation and results

In an earlier paper [14], we determined the ordinary and combinatorial depth of several subgroups of the simple Suzuki groups Sz.q/. Nontrivial subgroups having a disjoint conjugate are always of ordinary depth three in every group. We will show in Theorem 1.3 that, in the case of simple Suzuki groups, subgroups of ordinary depth 3 are exactly those nontrivial subgroups that have a disjoint conjugate. In Theorem 1.1, we characterize those nontrivial TI subgroups of Sz.q/ which are non-cyclic elementary abelian 2-subgroups These are exactly those subgroups that are conjugate to the center of a Sylow 2-subgroup of a smaller Suzuki subgroup Sz.s/ Ä Sz.q/. (iii) An elementary abelian subgroup K2n of order 2n > 2 is a TI-subgroup if and only if it is the center of a Sylow 2-subgroup of a simple Suzuki subgroup G1 Ä G, or a conjugate to it. Subgroups of ordinary depth 3 of a simple Suzuki group G D Sz.q/ are the following.

It is easy to see that

Preliminaries

Findings

Final remarks