Volume 14, Issue 3

In this paper we study Lyndon's equation x p y q z r = 1, with x, y, z group elements and p, q, r positive integers, in HNN extensions of free and fully residually free groups, and draw some conclusions about its behavior in Λ-free groups.

Triple factorizations of groups G of the form G = ABA , for proper subgroups A and B , are fundamental in the study of Lie type groups, as well as in geometry. They correspond to flag-transitive point-line incidence geometries in which each pair of points is incident with at least one line. This paper introduces and develops a general framework for studying triple factorizations of this form for finite groups, especially nondegenerate ones where G ≠ AB . We identify two necessary and suffcient conditions for subgroups A, B to satisfy G = ABA , in terms of the G -actions on the A -cosets and the B -cosets. This leads to an order (upper) bound for | G | in terms of | A | and | B | which is sharp precisely for the point-line incidence geometries of flag-transitive projective planes. We study in particular the case where G acts imprimitively on the A -cosets, inducing a permutation group that is naturally embedded in a wreath product G 0 ≀ G 1 . This gives rise to triple factorizations for G 0 , G 1 and G 0 ≀ G 1 , respectively. We present a rationale for further study of triple factorizations G = ABA in which A is maximal in G , and both A and B are core-free.

Let G be the group of order-preserving automorphisms of the rationals ℚ, or the group of colour-preserving automorphisms of the -coloured random graph . We show that given any non-identity ƒ ∈ G , there exists g ∈ G such that every automorphism in G is the limit of a sequence of automorphisms generated by ƒ and g . Moreover, if, in some sense, ƒ has no finite structure, then g can be chosen with a great deal of flexibility.

Let D be a defect group of a 2-block B of a finite group G . We conjecture that if D is a rational group and D ′ ⩽ Z ( D ), then the values of all χ ∈ Irr( B ) lie in a cyclotomic field ℚ m , for some odd integer m . We prove the conjecture when G is solvable or | D | = 8. Examples show that the condition D ′ ⩽ Z ( D ) cannot be relaxed.

Using combinatorics and character theory, we determine the imprimitive faithful complex characters, i.e., the irreducible faithful complex characters which are induced from proper subgroups, of the Schur covers of the symmetric and alternating groups. Furthermore, for every imprimitive character we establish all its minimal block stabilizers. As a corollary, we also determine the monomial faithful characters of the Schur covers.

We classify the real and strongly real conjugacy classes in GL n ( q ) and SL n ( q ). In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes. This paper is the first of two that together classify the real and strongly real classes in GL n ( q ), SL n ( q ), PGL n ( q ), PSL n ( q ), and all quasi-simple covers of PSL n ( q ).

We classify the real and strongly real conjugacy classes in PGL n ( q ), PSL n ( q ), and all quasi-simple covers of PSL n ( q ). In each case we give a formula for the number of real, and the number of strongly real, conjugacy classes. This is a companion paper to [Gill and Singh, J. Group Theory 14: 2011] in which we classified the real and strongly real conjugacy classes in GL n ( q ) and SL n ( q ).