Volume 17, Issue 4

Abstract. We address two questions of Simon Thomas. First, we show that for any n ≥ 3 one can find a four-generated free subgroup of SL n (ℤ) which is profinitely dense. More generally, we show that an arithmetic group Γ that admits the congruence subgroup property has a profinitely-dense free subgroup with an explicit bound on its rank. Next, we show that the set of profinitely-dense, locally-free subgroups of such an arithmetic group Γ is uncountable.

Abstract. We investigate the Jordan nilpotency of a group ring RG and, when RG has an involution that is the linear extension of an involution on G , also the Jordan nilpotency of the symmetric elements in RG .

Abstract. For k=1,2,...,n-1${k = 1, 2,\ldots ,n-1}$ let Vk=V(λk)${V_k = V(\lambda _k)}$ be the Weyl module for the special orthogonal group G= SO (2n+1,𝔽)${G = \mathrm {SO}(2n+1,\mathbb {F})}$ with respect to the k -th fundamental dominant weight λ k of the root system of type B n and put Vn=V(2λn)${V_n = V(2\lambda _n)}$. It is well known that all of these modules are irreducible when char(𝔽) ≠ 2 while when char(𝔽) = 2 they admit many proper submodules. In this paper, assuming that char(𝔽) = 2, we prove that V k admits a chain of submodules Vk=Mk⊃Mk-1⊃⋯⊃M1⊃M0⊃M-1=0${V_k = M_k \supset M_{k-1}\supset \dots \supset M_1\supset M_0 \supset M_{-1} = 0}$ where M i ≅ V i for 1,...,k-1${1,\ldots , k-1}$ and M 0 is the trivial 1-dimensional module. We also show that for i=1,2,...,k${i = 1, 2,\ldots , k}$ the quotient M i / M i -2 is isomorphic to the so-called i -th Grassmann module for G . Resting on this fact we can give a geometric description of M i -1 / M i -2 as a submodule of the i -th Grassmann module. When 𝔽 is perfect, G ≅ Sp(2 n ,𝔽) and M i / M i -1 is isomorphic to the Weyl module for Sp(2 n ,𝔽) relative to the i -th fundamental dominant weight of the root system of type C n . All irreducible sections of the latter modules are known. Thus, when 𝔽 is perfect, all irreducible sections of V k are known as well.

Abstract. We classify all strongly real conjugacy classes of the finite unitary group U( n ,𝔽 q ) when q is odd. In particular, we show that g ∈ U( n ,𝔽 q ) is strongly real if and only if g is an element of some embedded orthogonal group O ± ( n ,𝔽 q ). Equivalently, g is strongly real in U( n ,𝔽 q ) if and only if g is real and every elementary divisor of g of the form ( t ± 1) 2 m has even multiplicity. We apply this to obtain partial results on strongly real classes in the finite symplectic group Sp(2 n ,𝔽 q ), q odd, and a generating function for the number of strongly real classes in U( n ,𝔽 q ), q odd, and we also give partial results on strongly real classes in U( n ,𝔽 q ) when q is even.

Abstract. Let g be a random element of a finite classical group G , and let λ z -1 ( g ) denote the partition corresponding to the polynomial z - 1 in the rational canonical form of g . As the rank of G tends to infinity, λ z -1 ( g ) tends to a partition distributed according to a Cohen–Lenstra type measure on partitions. We give sharp upper and lower bounds on the total variation distance between the random partition λ z -1 ( g ) and the Cohen–Lenstra type measure.

Abstract. In this paper, we consider groups where the only prime power that occurs as a character degree is 1. We prove that the nonabelian composition factors of such a group also have this property or are the sporadic simple group M 12 . We characterize these groups that have a normal nonabelian Sylow p -subgroup.

Abstract. Thévenaz [Arch. Math. (Basel) 52 (1989), no. 3, 209–211] made an interesting observation that the number of conjugacy classes of cyclic subgroups in a finite group G is equal to the rank of the matrix of the numbers of double cosets in G . We give another proof of this fact and present a fusion system version of it. In particular we use finite groups realizing the fusion system ℱ as in our previous work [Arch. Math. (Basel) 94 (2010), no. 5, 405–410].

Abstract. Carlitz proved that, for any prime power q > 2, the group of all permutations of 𝔽 q is generated by the permutations induced by degree-one polynomials and x q -2 . His proof relies on a remarkable polynomial which appears to have been found by magic. We show here that no magic is required: there is a straightforward way to produce a simple polynomial which has the same remarkable properties as the complicated polynomial in Carlitz's proof. We also identify the crucial subtlety which allows such simple polynomials to exist, and discuss some consequences.

Abstract. In this paper a classification is given for finite soluble groups in which the normalizer of every non-normal cyclic subgroup is a maximal subgroup.

Abstract. We classify the finite 2-groups that have the property that all their non-abelian subgroups have the same center.

Abstract. Let Δ(G)${\Delta (G)}$ denote the intersection of all non-normal maximal subgroups of a group G . We introduce the class of T 2 -groups which are defined as the groups G for which G/Δ(G)${G/\Delta (G)}$ is a T -group, that is, a group in which normality is a transitive relation. Several results concerning the class T 2 are discussed. In particular, if G is a solvable group, then Sylow permutability is a transitive relation in G if and only if every subgroup H of G is a T 2 -group such that the nilpotent residual of H is a Hall subgroup of H .