Volume 18, Issue 3

We determine all saturated fusion systems ℱ on many 2-groups P with exactly three involutions. For this we use Janko's classification of these groups ([Groups of Prime Power Order. Vol. 1, Walter de Gruyter, Berlin, 2008, Section 82]) as a guide. On the way we describe all such P which contain an ℱ-essential subgroup isomorphic to C2k×Q8${C_{2^k} \times Q_8}$ or to the Sylow 2-subgroup of PSU(3,4). Furthermore, we classify all saturated fusion systems on these P . This contributes to the outstanding classification of fusion systems on 2-groups of 2-rank 2.

When ( S ,ℱ,ℒ) is a p -local finite group and ( T ,ℰ,ℒ 0 ) is weakly normal in ( S ,ℱ,ℒ), we show that a definition of C S (ℰ) given by Aschbacher has a simple interpretation from which one can deduce existence and strong closure very easily. We also appeal to a result of Gross to give a new proof that there is a unique fusion system C ℱ (ℰ) on C S (ℰ).

We introduce twisted permutation codes , which are frequency permutation arrays analogous to repetition permutation codes, namely, codes obtained from the repetition construction applied to a permutation code. In particular, we show that a lower bound for the minimum distance of a twisted permutation code is the minimum distance of a repetition permutation code. We give examples where this bound is tight, but more importantly, we give examples of twisted permutation codes with minimum distance strictly greater than this lower bound.

The notions of quasi-finitely axiomatizable (QFA) groups and quasi-axiomatizable (QA) groups were introduced by A. Nies in 2003. We say that a finitely generated group G is QFA if there exists a first-order sentence ϕ that is satisfied by G such that each finitely generated group that satisfies ϕ is isomorphic to G . The aim of this paper is to further investigate properties of direct products with respect to these notions, thereby extending F. Oger's result for nilpotent-by-finite groups. More precisely, we study the relations between (1) G and H are QFA, and (2) G × H is QFA. We prove that (1) implies (2) under the additional condition that there exist bounds on the commutator width in the central extensions of the center Z ( H ) by G and Z ( G ) by H . We also prove that the converse is true if the groups are ℤ-replaceable, in particular if they are residually soluble. Thus (1) and (2) are equivalent, for instance, in the classes of finitely generated abelian-by-polycyclic or abelian-by-metanilpotent groups.

Linear and projective boundaries of Cayley graphs were introduced in [Glasg. Math. J. (2014), DOI 10.1017/S0017089514000512] as quasi-isometry invariant boundaries of finitely generated groups. They consist of forward orbits g ∞ = { g i : i ∈ ℕ}, or orbits g ±∞ = { g i : i ∈ ℤ}, respectively, of non-torsion elements g of the group G , where `sufficiently close' (forward) orbits become identified, together with a metric bounded by 1. We show that for all finitely generated groups, the distance between the antipodal points g ∞ and g -∞ in the linear boundary is bounded from below by 1/2 1/2 , and we give an example of a group which has two antipodal elements of distance at most (12/17) 1/2 < 1. Our example is a derivation of the Baumslag–Gersten group. We also exhibit a group with elements g and h such that g ∞ = h ∞ , but g -∞ ≠ h -∞ . Furthermore, we introduce a notion of average-case-distortion—called growth—and compute explicit positive lower bounds for distances between points g ∞ and h ∞ which are limits of group elements g and h with different growth.

The pro-isomorphic zeta function ζ Γ ∧ ( s ) of a torsion-free finitely generated nilpotent group Γ enumerates finite index subgroups Δ ≤ Γ such that Δ and Γ have isomorphic profinite completions. It admits an Euler product decomposition ζ Γ ∧ ( s ) = ∏ p ζ Γ, p ∧ ( s ) We manufacture the first example of a torsion-free finitely generated nilpotent group Γ such that the local Euler factors ζ Γ, p ∧ ( s ) do not satisfy functional equations on inverting p . The group Γ has nilpotency class 4 and Hirsch length 25. It is obtained, via the Malcev correspondence, from a ℤ-Lie lattice Λ with a suitable algebraic automorphism group 𝐀𝐮𝐭(Λ).