Volume 18, Issue 6

We give upper bounds to the number of n -dimensional irreducible complex representations of finite quasisimple groups belonging to different families of groups of Lie type. The bounds have the form c n s , where c and s are explicit positive constants that both depend on the family in question. From these bounds, we deduce a uniform bound of the form c n to the number of n -dimensional irreducible representations of all finite quasisimple groups of Lie type. Finally, an application of these results to counting conjugacy classes of maximal subgroups of Lie groups is discussed.

In this paper we first investigate the relationship between the absolute center L ( G ) and the Frattini subgroup Φ( G ) for a finite group G , and then we describe the structure of finite groups G satisfying L ( G ) ≤ Φ( G ). For example, we prove that a finite group G is an ℱ-group if and only if G / L ( G ) is an ℱ-group, where ℱ is a saturated formation containing Z 2 . Next we determine all finite groups X such that X / L ( X ) is isomorphic to Z p ω × Z p , where ω is a positive integer and Z p ω is a cyclic group of order p ω .

In this paper, we study finite groups with some complemented subgroups of prime orders. We prove the p -supersolvability of a group with complemented subgroups of order p , where p is the second smallest prime divisor of the group order.

We study homomorphisms from Out( F 3 ) to Out( F 5 ) and GL m (𝕂) for m ≤ 6, where 𝕂 is a field of characteristic other than 2 or 3. We conclude that all 𝕂-linear representations of dimension at most 6 of Out( F 3 ) factor through GL 3 (ℤ), and that all homomorphisms Out( F 3 ) → Out( F 5 ) have finite image.

We show that the (standard restricted) wreath product of groups is boundedly generated if and only if the bottom group is boundedly generated and the top group is finite.

For any conjugacy class 𝒞 in G = PSL 2 ( q ) we compute 𝒞 2 and discuss whether or not 𝒞 contains a triple of elements whose product is 1 which generate G . Moreover, we determine which elements in G can be written as a product of two conjugate elements that generate G .

We characterize Beauville surfaces of unmixed type with group either PSL 2 ( p e ) or PSL 2 ( p e ), thus extending previous results of Bauer, Catanese and Grunewald, Fuertes and Jones, and Penegini and the author.

A group G is said to have the property R ∞ if for any of its automorphisms, φ, the Reidemeister number R (φ) is infinite. We study the relationships between the R ∞ property and nonabelian cohomology, isogredience classes, and representation theory.