Volume 10, Issue 4

In 1878, Jordan showed that a finite subgroup of GL( n , ℂ) must possess an abelian normal subgroup whose index is bounded by a function of n alone. We will give the optimal bound for all n ; for n ⩾ 71, it is ( n + 1)!, afforded by the symmetric group S n +1 . We prove a ‘replacement theorem’ that enables us to study linear groups by breaking them down into individual primitive constituents and we give detailed information about the structure of the groups that achieve the optimal bounds, for every degree n . Our proof relies on known lower bounds for the degrees of faithful representations of each quasisimple group, depending on the classification of finite simple groups, through the use of the bounds for primitive groups that the author has previously obtained.

We provide two explicit matrices of orders 2 and 3 that generate the group GL(6,ℤ).

We prove that |Ω { k } ( P )| = [ P : P p k ] in a powerful p -group P .

It is shown that if G is a finite p -group of coclass 2, then the order of G divides the order of the automorphism group of G .

In Part I it was shown that if G is a p -group of class k , generated by elements of orders , then a necessary condition for the capability of G is that r > 1 and . It was also shown that when G is the k -nilpotent product of the cyclic groups generated by those elements and k = p = 2 or k < p , then the given conditions are also sufficient. We make a correction related to the small class case, and extend the sufficiency result to k = p for an arbitrary prime p .

We discuss finiteness properties of a profinite group G whose probabilistic zeta function P G ( s ) is rational. In particular we prove that if P G ( s ) is rational and G has a finite number of non-alternating and non-abelian composition factors in a given composition series, then G /Frat( G ) is finite.

In this note we will give an elementary proof of the existence of sharply transitive R -modules M over principal ideal domains R . An R -module is sharply transitive (or a UT-module) if its R -automorphism group acts sharply transitively on the pure elements of M . We will assume that M is torsion-free; thus pure elements are simply those elements divisible only by units of R in M . We provide examples of UT-modules of rank ⩽ , while the existence of UT-modules of rank ⩾ was shown recently in Göbel and Shelah [R. Göbel and S. Shelah. Uniquely transitive torsion-free abelian groups. In Rings, modules, algebras, and abelian groups , Lecture Notes in Pure and Applied Math. 236 (Marcel Dekker, 2004), pp. 271–290.] using the more complicated machinery of prediction principles. The existence of countable abelian UT-groups, which follows from this note, was left open in earlier works. Here we require and exploit the existence of algebraically independent elements over the base ring R . (Thus we will need | R | < .) First we will convert the UT problem on modules (as suggested in Herden [D. Herden. Uniquely transitive R -modules. Ph.D. thesis. University of Duisburg-Essen, Campus Essen (2005).]) into a problem on suitable R -algebras. This reduces its solution to a few simple steps and makes the proofs more transparent, requiring only basic results in module theory.

For an arbitrary group G , and a G -adapted ring R (for example, R = ℤ), let 𝒰 be the group of units of the group ring RG , and let Z ∞ (𝒰) denote the union of the terms of the upper central series of 𝒰, the elements of which are called hypercentral units. It is shown that Z ∞ (𝒰) ⩽ ( G ). As a consequence, hypercentral units commute with all unipotent elements, and if G has non-normal finite subgroups with R( G ) denoting their intersection, then [𝒰,Z ∞ (𝒰)] ⩽ R( G ). Further consequences are given as well as concrete examples.

We extend a result in [J. R. J. Groves and D. Kochloukova. Embedding properties of metabelian Lie algebras and metabelian discrete groups. J. London Math. Soc. (2) 73 (2006), 475–492.] which showed that for each m every finitely generated metabelian group G embeds in a quotient of a metabelian group of homological type FP m and furthermore that G embeds in a metabelian group of type FP 4 . More precisely, we show that for a fixed m every finitely generated metabelian group G embeds in a metabelian group of type FP m .

We establish conditions on the defining graph of a right-angled Coxeter group presentation which guarantee that the boundary of any CAT(0) space on which the group acts geometrically will be locally connected.

Let F 2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for two given elements u , v of F 2 , u and v are translation equivalent in F 2 , that is, whether or not u and v have the property that the cyclic length of ( u ) equals the cyclic length of ( v ) for every automorphism of F 2 . This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edn. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) , Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.] for the case of F 2 .