Volume 9, Issue 6

This paper is a sequel to [A. M. Cohen and L. Paris. On a theorem of Artin. J. Group Theory 6 (2003), 421–441.]. Let A be an Artin group, let W be its associated Coxeter group, and let CA be its associated coloured Artin group, that is, the kernel of the standard epimorphism μ : A → W . We determine the homomorphisms φ : A → W that satisfy Im φ · Z(W) = W , for A irreducible and of spherical type, and we prove that CA is a characteristic subgroup of A if A is of spherical type but not necessarily irreducible.

This is the third in a series of papers whose object is to show how cycle index methods for finite classical groups, developed by Fulman [Jason Fulman. Cycle indices for the classical groups. J. Group Theory 2 (1999), 251–289.], may be extended to other almost simple groups of classical type. In [John R. Britnell. Cyclic, separable and semisimple transformations in the special unitary groups over a finite field. J. Group Theory 9 (2006), 547–569.] we treated the special unitary groups, and in [John R. Britnell. Cyclic, separable and semisimple transformations in the finite conformal groups. J. Group Theory 9 (2006), 571–601.] the general symplectic and general orthogonal groups. In this paper we shall treat various subgroups of the general orthogonal group over a field of odd characteristic. We shall focus at first on Ω ± ( d, q ), the commutator subgroup of Ο ± ( d, q ). Subsequently we shall look at groups G in the range where Π is the group of non-zero scalars.

The purpose of this note is to describe when a subgroup of a finite soluble group is an injector of that group, without directly using Fitting sets.

Let G be a finite group. We show that when the conjugacy class sizes of G are {1, m , n , mn }, with m and n positive integers such that ( m , n ) = 1, then G is solvable. As a consequence, we obtain that G is nilpotent and that m = p a and n = q b for two primes p and q .

A proper short exact sequence in the category of locally compact abelian groups is said to be t -pure if φ(A) is a topologically pure subgroup of B , that is, if for all positive integers n . We establish conditions under which t -pure exact sequences split and determine those locally compact abelian groups K ⊕ D (where K is compactly generated and D is discrete) which are t -pure injective or t -pure projective. Calling the extension (*) almost pure if for all positive integers n , we obtain a complete description of the almost pure injectives and almost pure projectives in the category of locally compact abelian groups.

Let F n be a free group of rank n ≥ 2. Two elements g , h in F n are said to be translation equivalent in F n if the cyclic length of φ ( g ) equals the cyclic length of φ ( h ) for every automorphism φ of F n . Let F(a, b) be the free group generated by { a, b } and let w(a, b) be an arbitrary word in F(a, b) . We prove that w(g, h) and w(h, g) are translation equivalent in F n whenever g , h ∈ F n are translation equivalent in F n , and thereby give an affermative solution to problem F38b in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edition. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math . 296 (American Mathematical Society, 2002), pp. 1–38.].

We discuss the solubility of equations of the form w = g , where w is a word (an element of a free group F X ) and g is an element of a given group G . A word for which this equation is soluble for every g ∈ G is said to be universal for G . It is conjectured that a word is universal for the automorphism group of the random graph if and only if it cannot be written as a proper power, corresponding to the results of [Randall Dougherty and Jan Mycielski. Representations of infinite permutations by words (II). Proc. Amer. Math. Soc. 127 (1999), 2233–43.], [Roger C. Lyndon. Words and infinite permutations. In Mots , Lang. Raison Calc. (Hermès, 1990), pp. 143–152.], [Jan Mycielski. Representations of infinite permutations by words. Proc. Amer. Math. Soc . 100 (1987), 237–241.], where the same necessary and sufficient condition was established for infinite symmetric groups. We prove various special cases. A key ingredient is the use of ‘generic’ automorphisms, and elements which suitably approximate them, called ‘special’.