Volume 17, Issue 2

Abstract. Given a finite subgroup Γ${\Gamma }$ of 𝐒𝐋3ℂ$\mathbf {SL}_3\mathbb {C}$, we determine how an arbitrary finite dimensional irreducible representation of 𝐒𝐋3ℂ$\mathbf {SL}_3\mathbb {C}$ decomposes under the action of Γ${\Gamma }$. To the subgroup Γ${\Gamma }$ we associate a generalized McKay matrix CΓ${C_\Gamma }$. Then, generalizing a method used by B. Kostant for 𝐒𝐋2ℂ$\mathbf {SL}_2\mathbb {C}$, we decompose CΓ${C_\Gamma }$ as a sum of products of reflections associated to mutually orthogonal roots: this is a sort of algebraic McKay correspondence in dimension 3.

Abstract. For a finite group G with subgroup H , the Chermak–Delgado measure of H in G refers to |H||CG(H)|${\vert H \vert \,\vert C_G(H) \vert }$. The set of all subgroups with maximal Chermak–Delgado measure form a sublattice, 𝒞𝒟(G)$\mathcal {CD}(G)$, within the subgroup lattice of G . This paper examines conditions under which the Chermak–Delgado lattice is a chain of subgroups H0<H1<⋯<Hn${H_0 &lt; H_1 &lt; \cdots &lt; H_n}$. On the basis of a general result about extending certain Chermak–Delgado lattices, we construct, for any prime p and any non-negative integer n , a p -group whose Chermak–Delgado lattice is a chain of length n .

Abstract. Every finite p -group of coclass 2 has a noninner automorphism of order p leaving the center elementwise fixed.

Abstract. We describe finite soluble groups in which every n -maximal subgroup is 𝔉${\mathfrak {F}}$-subnormal.

Abstract. We enumerate and describe the path components for the spaces of ordered commuting k -tuples of orthogonal and special orthogonal matrices respectively. We provide a lower bound on the number of components for the spinor groups.

Abstract. We prove that if the probabilistic zeta function PG(s)${P_G(s)}$ of a finitely generated profinite group G is rational and all but finitely many nonabelian composition factors of G are groups of Lie type in a fixed characteristic or sporadic simple groups, then G contains only finitely many maximal subgroups.

Abstract. A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We construct the JSJ decomposition of a vGBS group over abelian groups. We prove that this decomposition is explicitly computable, and may be obtained by local changes on the initial graph of groups.

Abstract. We construct a class of finitely presented groups within which the isomorphism problem is solvable but the commensurability problem is unsolvable. Conversely, we construct a class of finitely presented groups within which the commensurability problem is solvable but the isomorphism problem is unsolvable. These are the first examples of such a contrastive complexity behavior with respect to the isomorphism problem.

Abstract. Corrigendum to http://dx.doi.org/10.1515/jgt.2010.053.