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Abstract We study certain one-parameter families of exponential sums of Airy–Laurent type. Their general theory was developed in Katz and Tiep (Airy sheaves of Laurent type: an introduction,https://web.math.princeton.edu/~nmk/kt31_11sept.pdf). In the present paper, we make use of that general theory to compute monodromy groups in some particularly simple families (in the sense of “simple to remember), realizing Weyl groups of type$$E_6$$ $E_6$and$$E_8$$ $E_8$. 

Abstract Letk(B₀) andl(B₀) respectively denote the number of ordinary andp-Brauer irreducible characters in the principal blockB₀of a finite groupG. We prove that, ifk(B₀)−l(B₀) = 1, thenl(B₀) ≥p− 1 or elsep= 11 andl(B₀) = 9. This follows from a more general result that for every finite groupGin which all non-trivialp-elements are conjugate,l(B₀) ≥p− 1 or elsep= 11 and$$G/{{\bf{O}}_{{p^\prime }}}(G) \cong C_{11}^2\, \rtimes\,{\rm{SL}}(2,5)$$ $G/O_{p^{\prime }}\left(G\right)\cong C_{11}^2⋊\text{SL}(2,5)$. These results are useful in the study of principal blocks with few characters. We propose that, in every finite groupGof order divisible byp, the number of irreducible Brauer characters in the principalp-block ofGis always at least$$2\sqrt {p - 1} + 1 - {k_p}(G)$$ $2\sqrt{p-1}+1-k_p\left(G\right)$, wherek_p(G) is the number of conjugacy classes ofp-elements ofG. This indeed is a consequence of the celebrated Alperin weight conjecture and known results on bounding the number ofp-regular classes in finite groups. 

Abstract The main results of the paper develop a level theory and establish strong character bounds for finite classical groups, in the case that the centralizer of the element has small order compared to$$|G|$$ $\left|G\right|$in a logarithmic sense. 

Abstract We prove that the number of conjugacy classes of a finite groupGconsisting of elements of odd order, is larger than or equal to that number for the normaliser of a Sylow 2-subgroup ofG. This is predicted by the Alperin Weight Conjecture. 

We completely describe all the possible fields of values of irreducible characters of degree up to 3 of finite groups. The obtained result points toward a rather surprising connection between the field of values and the degree of an arbitrary irreducible character. 

We prove that if G is a nonsolvable group, then the proportion of vanishing elements of G is at least 1067/1260 (and this lower bound is optimal). This confirms a conjecture of Dolfi, Pacifici, and Sanus [7]. 

Abstract For every integer k there exists a bound $$B=B(k)$$ B = B ( k ) such that if the characteristic polynomial of $$g\in \textrm{SL}_n(q)$$ g ∈ SL n ( q ) is the product of $$\le k$$ ≤ k pairwise distinct monic irreducible polynomials over $$\mathbb {F}_q$$ F q , then every element x of $$\textrm{SL}_n(q)$$ SL n ( q ) of support at least B is the product of two conjugates of g . We prove this and analogous results for the other classical groups over finite fields; in the orthogonal and symplectic cases, the result is slightly weaker. With finitely many exceptions ( p ,  q ), in the special case that $$n=p$$ n = p is prime, if g has order $$\frac{q^p-1}{q-1}$$ q p - 1 q - 1 , then every non-scalar element $$x \in \textrm{SL}_p(q)$$ x ∈ SL p ( q ) is the product of two conjugates of g . The proofs use the Frobenius formula together with upper bounds for values of unipotent and quadratic unipotent characters in finite classical groups.