Volume 25, Issue 4

The subgroup zeta function and the normal zeta function of a finitely generated virtually nilpotent group can be expressed as finite sums of Dirichlet series admitting Euler product factorization. We compute these series except for a finite number of local factors when the group is virtually nilpotent of Hirsch length 3. We deduce that they can be meromorphically continued to the whole complex plane and that they satisfy local functional equations. The complete computation (with no exception of local factors) is presented for those groups that are also torsion-free, that is, for the 3-dimensional almost-Bieberbach groups.

In the present paper, we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations B n → GL n ⁢ ( n - 1 ) / 2 ⁢ ( Z ⁢ [ t ± 1 ] ) B_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1}]) , VB n → GL n ⁢ ( n - 1 ) / 2 ⁢ ( Z ⁢ [ t ± 1 , t 1 ± 1 , t 2 ± 1 , … , t n - 1 ± 1 ] ) \mathrm{VB}_{n}\to\mathrm{GL}_{n(n-1)/2}(\mathbb{Z}[t^{\pm 1},t_{1}^{\pm 1},t_{2}^{\pm 1},\ldots,t_{n-1}^{\pm 1}]) which are connected with the famous Lawrence–Bigelow–Krammer representation. It turns out that these representations induce faithful representations of the crystallographic groups B n / P n ′ B_{n}/P_{n}^{\prime} , VB n / VP n ′ \mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime} , respectively. Using these representations we study certain properties of the groups B n / P n ′ B_{n}/P_{n}^{\prime} , VB n / VP n ′ \mathrm{VB}_{n}/\mathrm{VP}_{n}^{\prime} . Moreover, we construct new representations and decompositions of the universal braid groups UB n \mathrm{UB}_{n} .

We show that if a nilpotent group 𝐺 has a balanced presentation and Hirsch length h ⁢ ( G ) > 3 h(G)>3 , then β 1 ⁢ ( G ; Q ) = 2 \beta_{1}(G;\mathbb{Q})=2 . There is one such group which is torsion-free and of Hirsch length h = 4 h=4 , and none with h = 5 h=5 . We also construct a torsion-free nilpotent group 𝐺 with h ⁢ ( G ) = 6 h(G)=6 and β 2 ⁢ ( G ; F ) = β 1 ⁢ ( G ; F ) \beta_{2}(G;F)=\beta_{1}(G;F) for all fields 𝐹.

For a character 𝜒 of a finite group 𝐺, the number cod ( χ ) := | G : ker χ | / χ ( 1 ) \operatorname{cod}(\chi):=\lvert G:\operatorname{ker}\chi\rvert/\chi(1) is called the codegree of 𝜒. The object of this paper is to classify the finite groups 𝐺 which have exactly one irreducible character 𝜒 such that χ ⁢ ( 1 ) ∤ cod ⁡ ( χ ) \chi(1)\nmid\operatorname{cod}(\chi) .

We present a description of non-solvable groups in which all real irreducible character degrees are prime-power numbers.

A Hurwitz generating triple for a group 𝐺 is an ordered triple of elements ( x , y , z ) ∈ G 3 (x,y,z)\in G^{3} , where x 2 = y 3 = z 7 = x ⁢ y ⁢ z = 1 x^{2}=y^{3}=z^{7}=xyz=1 and ⟨ x , y , z ⟩ = G \langle x,y,z\rangle=G . For the finite quasisimple exceptional groups of types F 4 F_{4} , E 6 E_{6} , E 6 2 {}^{2}E_{6} , E 7 E_{7} and E 8 E_{8} , we provide restrictions on which conjugacy classes 𝑥, 𝑦 and 𝑧 can belong to if ( x , y , z ) (x,y,z) is a Hurwitz generating triple. We prove that there exist Hurwitz generating triples for F 4 ⁢ ( 3 ) F_{4}(3) , F 4 ⁢ ( 5 ) F_{4}(5) , F 4 ⁢ ( 7 ) F_{4}(7) , F 4 ⁢ ( 8 ) F_{4}(8) , E 6 ⁢ ( 3 ) E_{6}(3) and E 7 ⁢ ( 2 ) E_{7}(2) , and that there are no such triples for F 4 ⁢ ( 2 3 ⁢ n - 2 ) F_{4}(2^{3n-2}) , F 4 ⁢ ( 2 3 ⁢ n - 1 ) F_{4}(2^{3n-1}) , E 6 ⁢ ( 7 3 ⁢ n - 2 ) E_{6}(7^{3n-2}) , E 6 ⁢ ( 7 3 ⁢ n - 1 ) E_{6}(7^{3n-1}) , S ⁢ E 6 ⁢ ( 7 n ) SE_{6}(7^{n}) or E 6 2 ⁢ ( 7 n ) {}^{2}E_{6}(7^{n}) when n ≥ 1 n\geq 1 .

By using transversals belonging to the commutant, we characterize Moufang loops whose commutant is a normal subloop, i.e. we give necessary and sufficient conditions in the multiplication group for this purpose.