Strongly real groups and totally orthogonal groups form two important subclasses of real groups. In this article we give a characterization of strongly real special 2-groups. This characterization is in terms of quadratic maps over fields of characteristic 2. We then provide examples of groups which are in one subclass and not the other. It is a conjecture of Tiep that such examples are not possible for finite simple groups. Like O n (q) there are plenty of groups which are both totally orthogonal and strongly real. According to a conjecture of Tiep finite simple groups are totally orthogonal if and only if they 2000 Mathematics Subject Classification. 20C15, 20C33.
Let 𝐺 be a connected reductive group defined over F q \mathbb{F}_{q} . Fix an integer M ≥ 2 M\geq 2 , and consider the power map x ↦ x M x\mapsto x^{M} on 𝐺. We denote the image of G ( F q ) G(\mathbb{F}_{q}) under this map by G ( F q ) M G(\mathbb{F}_{q})^{M} and estimate what proportion of regular semisimple, semisimple and regular elements of G ( F q ) G(\mathbb{F}_{q}) it contains. We prove that, as q → ∞ q\to\infty , the set of limits for each of these proportions is the same and provide a formula. This generalizes the well-known results for M = 1 M=1 where all the limits take the same value 1. We also compute this more explicitly for the groups GL ( n , q ) \mathrm{GL}(n,q) and U ( n , q ) \mathrm{U}(n,q) and show that the set of limits are the same for these two group, in fact, in bijection under q ↦ - q q\mapsto-q for a fixed 𝑀.
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