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.
For a natural number m, a Lie algebra L over a field k is said to be of breadth type (0, m) if the co-dimension of the centralizer of every non-central element is of dimension m. In this article, we classify finite dimensional nilpotent Lie algebras of breadth type (0, 3) over Fq of odd characteristics up to isomorphism. We also give a partial classification of the same over finite fields of even characteristic, C and R. We also discuss 2-step nilpotent Camina Lie algebras.
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Consider the set of all powers GL(n, q) M = {x M | x ∈ GL(n, q)} for an integer M ≥ 2. In this article, we aim to enumerate the regular, regular semisimple and semisimple elements as well as conjugacy classes in the set GL(n, q) M , i.e., the elements or classes of these kinds which are M th powers. We get the generating functions for (i) regular and regular semisimple elements (and classes) when (q, M ) = 1, (ii) for semisimple elements and all elements (and classes) when M is a prime power and (q, M ) = 1, and (iii) for all kinds when M is a prime and q is a power of M .
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